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kernel

Selects and applies a specified kernel function.

Parameters:

Name Type Description Default
kernel_name str

The name of the kernel to apply. Supported kernels include: - 'linear_kernel', 'polynomial_kernel', 'hyperbolic_tangent_kernel' - 'exponential_kernel', 'cosine_similarity_kernel', 'euclidean_distance' - 'minkowski_distance', 'manhattan_distance', 'chebyshev_distance' - 'canberra_distance', 'gaussian_rbf_kernel', 'laplacian_kernel', - 'anisotropic_rbf_kernel', 'custom_hybrid_kernel'.

 required
x Tensor

Input tensor. Required for most kernels.

 None
x2 Tensor

Secondary input tensor. Required for non-batch kernels.

 None

Returns:

Type Description
Tensor

The result of the selected kernel function.

Raises:

Type Description
ValueError

If the input tensors are invalid.

Examples:

>>> kernel('linear_kernel', x=torch.tensor([1.0, 2.0]), x2=torch.tensor([3.0, 4.0]))
tensor(11.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def kernel(
    kernel_name: str,
    x: torch.Tensor = None,
    x2: torch.Tensor = None,
    *args, **kwargs
):
    """
        Selects and applies a specified kernel function.

        Parameters
        ----------
        kernel_name : str
            The name of the kernel to apply. Supported kernels include:
            - 'linear_kernel', 'polynomial_kernel', 'hyperbolic_tangent_kernel'
            - 'exponential_kernel', 'cosine_similarity_kernel', 'euclidean_distance'
            - 'minkowski_distance', 'manhattan_distance', 'chebyshev_distance'
            - 'canberra_distance', 'gaussian_rbf_kernel', 'laplacian_kernel',
            - 'anisotropic_rbf_kernel', 'custom_hybrid_kernel'.
        x : torch.Tensor, optional
            Input tensor. Required for most kernels.
        x2 : torch.Tensor, optional
            Secondary input tensor. Required for non-batch kernels.

        Returns
        -------
        torch.Tensor
            The result of the selected kernel function.

        Raises
        ------
        ValueError
            If the input tensors are invalid.

        Examples
        --------
        >>> kernel('linear_kernel', x=torch.tensor([1.0, 2.0]), x2=torch.tensor([3.0, 4.0]))
        tensor(11.)
    """

    if 'batch' in kernel_name:
        assert x is not None and x2 is None
    else:
        assert x is not None and x2 is not None

    match kernel_name:
        case 'linear_kernel' | 'inner_product' | 'dot_product': return linear_kernel(x=x, x2=x2, *args, **kwargs)
        case 'polynomial_kernel' | 'polynomial': return polynomial_kernel(x=x, x2=x2, *args, **kwargs)
        case 'hyperbolic_tangent_kernel' | 'hyperbolic_tangent': return hyperbolic_tangent_kernel(x=x, x2=x2, *args, **kwargs)
        case 'exponential_kernel' | 'exponential': return exponential_kernel(x=x, x2=x2, *args, **kwargs)
        case 'cosine_similarity_kernel' | 'cosine_similarity': return cosine_similarity_kernel(x=x, x2=x2, *args, **kwargs)
        case 'euclidean_distance': return euclidean_distance(x=x, x2=x2, *args, **kwargs)
        case 'minkowski_distance': return minkowski_distance(x=x, x2=x2, *args, **kwargs)
        case 'manhattan_distance': return manhattan_distance(x=x, x2=x2, *args, **kwargs)
        case 'chebyshev_distance': return chebyshev_distance(x=x, x2=x2, *args, **kwargs)
        case 'canberra_distance': return canberra_distance(x=x, x2=x2, *args, **kwargs)
        case 'euclidean_distance_kernel': return euclidean_distance_kernel(x=x, x2=x2, *args, **kwargs)
        case 'minkowski_distance_kernel': return minkowski_distance_kernel(x=x, x2=x2, *args, **kwargs)
        case 'manhattan_distance_kernel': return manhattan_distance_kernel(x=x, x2=x2, *args, **kwargs)
        case 'chebyshev_distance_kernel': return chebyshev_distance_kernel(x=x, x2=x2, *args, **kwargs)
        case 'canberra_distance_kernel': return canberra_distance_kernel(x=x, x2=x2, *args, **kwargs)
        case 'gaussian_rbf_kernel' | 'gaussian_rbf': return gaussian_rbf_kernel(x=x, x2=x2, *args, **kwargs)
        case 'laplacian_kernel' | 'laplacian': return laplacian_kernel(x=x, x2=x2, *args, **kwargs)
        case 'anisotropic_rbf_kernel' | 'anisotropic_rbf' | 'anisotropic': return anisotropic_rbf_kernel(x=x, x2=x2, *args, **kwargs)
        case 'custom_hybrid_kernel' | 'custom_hybrid' | 'custom': return custom_hybrid_kernel(x=x, x2=x2, *args, **kwargs)

Computes the linear kernel (inner product) between tensors.

Parameters:

Name Type Description Default
x Tensor

The input tensor.

 required
x2 Tensor

Secondary input tensor for pairwise kernel computation.

 None
centered bool

Whether to center the data before computing the kernel.

 False
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The kernel matrix or scalar value.

Raises:

Type Description
ValueError

If input tensors are invalid.

Examples:

>>> linear_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[ 5., 11.],
        [11., 25.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def linear_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the linear kernel (inner product) between tensors.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor.
        x2 : torch.Tensor, optional
            Secondary input tensor for pairwise kernel computation.
        centered : bool, default=False
            Whether to center the data before computing the kernel.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The kernel matrix or scalar value.

        Raises
        ------
        ValueError
            If input tensors are invalid.

        Examples
        --------
        >>> linear_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[ 5., 11.],
                [11., 25.]])
    """
    if x2 is None:
        return batch_linear_kernel(x, centered=centered, dim=dim)
    else:
        return instance_linear_kernel(x1=x, x2=x2)

Computes the dot product between two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required

Returns:

Type Description
Tensor

The scalar dot product.

Raises:

Type Description
ValueError

If tensors are invalid or have mismatched dimensions.

Examples:

>>> instance_linear_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(11.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_linear_kernel(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the dot product between two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.

        Returns
        -------
        torch.Tensor
            The scalar dot product.

        Raises
        ------
        ValueError
            If tensors are invalid or have mismatched dimensions.

        Examples
        --------
        >>> instance_linear_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(11.)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')
    dot_product = torch.dot(x1, x2)
    return dot_product

Computes the batch linear kernel matrix for a 2D tensor.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
centered bool

Whether to center the data.

 False
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The linear kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If the input tensor is invalid.

Examples:

>>> batch_linear_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[ 5., 11.],
        [11., 25.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_linear_kernel(x: torch.Tensor, centered: bool = False, dim: int = 0):
    """
        Computes the batch linear kernel matrix for a 2D tensor.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The linear kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If the input tensor is invalid.

        Examples
        --------
        >>> batch_linear_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[ 5., 11.],
                [11., 25.]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if dim == 1:
        x = x.T
    b, m = x.shape

    if centered:
        x = x - torch.mean(x, dim=0, keepdim=True)
    inner_product_matrix = torch.matmul(x.T, x)

    assert inner_product_matrix.shape == (m, m)
    return inner_product_matrix

Computes the polynomial kernel between tensors.

Parameters:

Name Type Description Default
x Tensor

Input tensor.

 required
x2 Tensor

Secondary input tensor for pairwise computation.

 None
centered bool

Whether to center the data.

 False
c float

Constant term in the kernel.

 0.0
d int

Degree of the polynomial.

 1
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The polynomial kernel matrix.

Raises:

Type Description
ValueError

If the inputs are invalid or incompatible.

Examples:

>>> polynomial_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), d=2, dim=1)
tensor([[ 25., 121.],
        [121., 625.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def polynomial_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, c: float = 0.0, d: int = 1, dim: int = 0):
    """
        Computes the polynomial kernel between tensors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor.
        x2 : torch.Tensor, optional
            Secondary input tensor for pairwise computation.
        centered : bool, default=False
            Whether to center the data.
        c : float, default=0.0
            Constant term in the kernel.
        d : int, default=1
            Degree of the polynomial.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The polynomial kernel matrix.

        Raises
        ------
        ValueError
            If the inputs are invalid or incompatible.

        Examples
        --------
        >>> polynomial_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), d=2, dim=1)
        tensor([[ 25., 121.],
                [121., 625.]])
    """
    if x2 is None:
        return batch_polynomial_kernel(x=x, centered=centered, c=c, d=d, dim=dim)
    else:
        return instance_polynomial_kernel(x1=x, x2=x2, c=c, d=d)

Computes the polynomial kernel for two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
c float

Constant term in the kernel.

 0.0
d int

Degree of the polynomial.

 1

Returns:

Type Description
Tensor

The scalar polynomial kernel value.

Raises:

Type Description
ValueError

If the inputs are invalid or incompatible.

Examples:

>>> instance_polynomial_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), d=2)
tensor(121.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_polynomial_kernel(x1: torch.Tensor, x2: torch.Tensor, c: float = 0.0, d: int = 1):
    """
        Computes the polynomial kernel for two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        c : float, default=0.0
            Constant term in the kernel.
        d : int, default=1
            Degree of the polynomial.

        Returns
        -------
        torch.Tensor
            The scalar polynomial kernel value.

        Raises
        ------
        ValueError
            If the inputs are invalid or incompatible.

        Examples
        --------
        >>> instance_polynomial_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), d=2)
        tensor(121.)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')

    dot_product = torch.dot(x1, x2)

    if dot_product + c == 0 and d < 0:
        raise ValueError("The negative powers of zeros is invalid...")

    return (dot_product + c)**d

Computes the batch polynomial kernel matrix for a 2D tensor.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
centered bool

Whether to center the data.

 False
c float

Constant term in the kernel.

 0.0
d int

Degree of the polynomial.

 1
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The polynomial kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If the input tensor is invalid.

Examples:

>>> batch_polynomial_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), d=2, dim=1)
tensor([[ 25., 121.],
        [121., 625.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_polynomial_kernel(x: torch.Tensor, centered: bool = False, c: float = 0.0, d: int = 1, dim: int = 0):
    """
        Computes the batch polynomial kernel matrix for a 2D tensor.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        centered : bool, default=False
            Whether to center the data.
        c : float, default=0.0
            Constant term in the kernel.
        d : int, default=1
            Degree of the polynomial.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The polynomial kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If the input tensor is invalid.

        Examples
        --------
        >>> batch_polynomial_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), d=2, dim=1)
        tensor([[ 25., 121.],
                [121., 625.]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if dim == 1:
        x = x.T
    b, m = x.shape

    if centered:
        x = x - torch.mean(x, dim=0, keepdim=True)
    inner_product_matrix = torch.matmul(x.T, x)

    if torch.any(inner_product_matrix + c == 0) and d < 0:
        raise ValueError("The matrix contains zero elements and the negative powers of zeros is invalid...")

    assert inner_product_matrix.shape == (m, m)
    return (inner_product_matrix + c)**d

Computes the hyperbolic tangent kernel for tensors.

Parameters:

Name Type Description Default
x Tensor

Input tensor.

 required
x2 Tensor

Secondary input tensor for pairwise computation.

 None
centered bool

Whether to center the data.

 False
c float

Bias term in the kernel.

 0.0
alpha float

Slope of the activation function.

 1.0
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The hyperbolic tangent kernel matrix or scalar value.

Examples:

>>> hyperbolic_tangent_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]))
tensor([[1., 1.],
        [1., 1.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def hyperbolic_tangent_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, c: float = 0.0, alpha: float = 1.0, dim: int = 0):
    """
        Computes the hyperbolic tangent kernel for tensors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor.
        x2 : torch.Tensor, optional
            Secondary input tensor for pairwise computation.
        centered : bool, default=False
            Whether to center the data.
        c : float, default=0.0
            Bias term in the kernel.
        alpha : float, default=1.0
            Slope of the activation function.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The hyperbolic tangent kernel matrix or scalar value.

        Examples
        --------
        >>> hyperbolic_tangent_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]))
        tensor([[1., 1.],
                [1., 1.]])
    """
    if x2 is None:
        return batch_hyperbolic_tangent_kernel(x=x, centered=centered, c=c, alpha=alpha, dim=dim)
    else:
        return instance_hyperbolic_tangent_kernel(x1=x, x2=x2, c=c, alpha=alpha)

Computes the hyperbolic tangent kernel for two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
c float

Bias term in the kernel.

 0.0
alpha float

Slope of the activation function.

 1.0

Returns:

Type Description
Tensor

The scalar kernel value.

Examples:

>>> instance_hyperbolic_tangent_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(1.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_hyperbolic_tangent_kernel(x1: torch.Tensor, x2: torch.Tensor, c: float = 0.0, alpha: float = 1.0):
    """
        Computes the hyperbolic tangent kernel for two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        c : float, default=0.0
            Bias term in the kernel.
        alpha : float, default=1.0
            Slope of the activation function.

        Returns
        -------
        torch.Tensor
            The scalar kernel value.

        Examples
        --------
        >>> instance_hyperbolic_tangent_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(1.)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')

    dot_product = torch.dot(x1, x2)

    return torch.tanh(alpha*dot_product + c)

Computes the batch hyperbolic tangent kernel matrix for a 2D tensor.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
centered bool

Whether to center the data.

 False
c float

Bias term in the kernel.

 0.0
alpha float

Slope of the activation function.

 1.0
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The kernel matrix of shape (m, m).

Examples:

>>> batch_hyperbolic_tangent_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]))
tensor([[1., 1.],
        [1., 1.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_hyperbolic_tangent_kernel(x: torch.Tensor, centered: bool = False, c: float = 0.0, alpha: float = 1.0, dim: int = 0):
    """
        Computes the batch hyperbolic tangent kernel matrix for a 2D tensor.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        centered : bool, default=False
            Whether to center the data.
        c : float, default=0.0
            Bias term in the kernel.
        alpha : float, default=1.0
            Slope of the activation function.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The kernel matrix of shape (m, m).

        Examples
        --------
        >>> batch_hyperbolic_tangent_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]))
        tensor([[1., 1.],
                [1., 1.]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if dim == 1:
        x = x.T
    b, m = x.shape

    if centered:
        x = x - torch.mean(x, dim=0, keepdim=True)
    inner_product_matrix = torch.matmul(x.T, x)

    assert inner_product_matrix.shape == (m, m)
    return torch.tanh(alpha*inner_product_matrix + c)

Computes the cosine similarity kernel for tensors.

Parameters:

Name Type Description Default
x Tensor

Input tensor.

 required
x2 Tensor

Secondary input tensor for pairwise computation.

 None
centered bool

Whether to center the data.

 False
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The cosine similarity matrix or scalar value.

Examples:

>>> cosine_similarity_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.9839],
        [0.9839, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def cosine_similarity_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the cosine similarity kernel for tensors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor.
        x2 : torch.Tensor, optional
            Secondary input tensor for pairwise computation.
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The cosine similarity matrix or scalar value.

        Examples
        --------
        >>> cosine_similarity_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.9839],
                [0.9839, 1.0000]])
    """
    if x2 is None:
        return batch_cosine_similarity_kernel(x=x, centered=centered, dim=dim)
    else:
        return instance_cosine_similarity_kernel(x1=x, x2=x2)

Computes the cosine similarity between two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required

Returns:

Type Description
Tensor

The scalar similarity value.

Examples:

>>> instance_cosine_similarity_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.9839)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_cosine_similarity_kernel(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the cosine similarity between two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.

        Returns
        -------
        torch.Tensor
            The scalar similarity value.

        Examples
        --------
        >>> instance_cosine_similarity_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.9839)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')

    dot_product = torch.dot(x1, x2)
    norm_x1 = torch.norm(x1, p=2)
    norm_x2 = torch.norm(x2, p=2)

    if norm_x1 == 0 or norm_x2 == 0:
        return torch.tensor(0.0)
    else:
        cosine_sim = dot_product / (norm_x1 * norm_x2)
        return cosine_sim

Computes the batch cosine similarity kernel matrix for a 2D tensor.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
centered bool

Whether to center the data.

 False
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The cosine similarity matrix of shape (m, m).

Examples:

>>> batch_cosine_similarity_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.9839],
        [0.9839, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_cosine_similarity_kernel(x: torch.Tensor, centered: bool = False, dim: int = 0):
    """
        Computes the batch cosine similarity kernel matrix for a 2D tensor.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The cosine similarity matrix of shape (m, m).

        Examples
        --------
        >>> batch_cosine_similarity_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.9839],
                [0.9839, 1.0000]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if dim == 1:
        x = x.T
    b, m = x.shape

    if centered:
        x = x - torch.mean(x, dim=0, keepdim=True)
    x_norm = torch.norm(x, p=2, dim=0, keepdim=True)
    x_norm[x_norm == 0] = torch.tensor(1.0)
    x_normalized = x / x_norm
    similarity_matrix = torch.matmul(x_normalized.T, x_normalized)

    assert similarity_matrix.shape == (m, m)
    return similarity_matrix

Computes the Minkowski distance kernel.

Parameters:

Name Type Description Default
x Tensor

Input tensor.

 required
x2 Tensor

Secondary input tensor for pairwise computation.

 None
p Union[int, float, str, Any]

Order of the Minkowski distance.

 None
centered bool

Whether to center the data.

 False
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The distance kernel matrix or scalar value.

Examples:

>>> minkowski_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
tensor([[1.0000, 0.0591],
        [0.0591, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def minkowski_distance_kernel(x: torch.Tensor, x2: torch.Tensor = None, p: Union[int, float, str, Any] = None, centered: bool = False, dim: int = 0):
    """
        Computes the Minkowski distance kernel.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor.
        x2 : torch.Tensor, optional
            Secondary input tensor for pairwise computation.
        p : Union[int, float, str, Any], optional
            Order of the Minkowski distance.
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The distance kernel matrix or scalar value.

        Examples
        --------
        >>> minkowski_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
        tensor([[1.0000, 0.0591],
                [0.0591, 1.0000]])
    """
    return torch.exp(-minkowski_distance(x=x, x2=x2, p=p, centered=centered, dim=dim))

Computes the Minkowski distance between tensors, either as a batch or for a single pair.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
p Union[int, float, str, Any]

The order of the Minkowski distance. Must be positive or a valid norm type (e.g., 'fro', 'nuc').

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed when in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar value representing the Minkowski distance.
  • For batch computation (x2 is None): A distance matrix of shape (m, m).

Examples:

Pairwise Minkowski distance:

>>> minkowski_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), p=2)
tensor(2.8284)

Batch Minkowski distance:

>>> minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
tensor([[0.0000, 2.8284],
        [2.8284, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def minkowski_distance(x: torch.Tensor, x2: torch.Tensor = None, p: Union[int, float, str, Any] = None, centered: bool = False, dim: int = 0):
    """
        Computes the Minkowski distance between tensors, either as a batch or for a single pair.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        p : Union[int, float, str, Any], optional
            The order of the Minkowski distance. Must be positive or a valid norm type (e.g., 'fro', 'nuc').
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed when in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar value representing the Minkowski distance.
            - For batch computation (x2 is None): A distance matrix of shape (m, m).

        Examples
        --------
        Pairwise Minkowski distance:
        >>> minkowski_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), p=2)
        tensor(2.8284)

        Batch Minkowski distance:
        >>> minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
        tensor([[0.0000, 2.8284],
                [2.8284, 0.0000]])
    """
    if x2 is None:
        return batch_minkowski_distance(x=x, p=p, centered=centered, dim=dim)
    else:
        return instance_minkowski_distance(x1=x, x2=x2, p=p)

Computes the Minkowski distance between two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
p Union[int, float, str, Any]

The order of the Minkowski distance. Must be positive or a valid norm type.

 None

Returns:

Type Description
Tensor

The Minkowski distance as a scalar value.

Examples:

>>> instance_minkowski_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), p=2)
tensor(2.8284)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_minkowski_distance(x1: torch.Tensor, x2: torch.Tensor, p: Union[int, float, str, Any] = None):
    """
        Computes the Minkowski distance between two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        p : Union[int, float, str, Any], optional
            The order of the Minkowski distance. Must be positive or a valid norm type.

        Returns
        -------
        torch.Tensor
            The Minkowski distance as a scalar value.

        Examples
        --------
        >>> instance_minkowski_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), p=2)
        tensor(2.8284)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')
    if p is None:
        raise ValueError('p must be provided and cannot be None...')
    if isinstance(p, str) and p not in ['fro', 'nuc']:
        raise ValueError('p must be either "fro" or "nuc" or torch.inf (numpy.inf) or -torch.inf (numpy.inf) or numbers...')

    distance = torch.norm(x1 - x2, p=p)

    return distance

Computes the Minkowski distance matrix for a batch of vectors.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
p Union[int, float, str, Any]

The order of the Minkowski distance. Must be positive or a valid norm type.

 None
centered bool

Whether to center the data.

 False
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A distance matrix of shape (m, m).

Examples:

>>> batch_minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2)
tensor([[0.0000, 1.4142],
        [1.4142, 0.0000]])

>>> batch_minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
tensor([[0.0000, 2.8284],
        [2.8284, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_minkowski_distance(x: torch.Tensor, p: Union[int, float, str, Any] = None, centered: bool = False, dim: int = 0):
    """
        Computes the Minkowski distance matrix for a batch of vectors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        p : Union[int, float, str, Any], optional
            The order of the Minkowski distance. Must be positive or a valid norm type.
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A distance matrix of shape (m, m).

        Examples
        --------
        >>> batch_minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2)
        tensor([[0.0000, 1.4142],
                [1.4142, 0.0000]])

        >>> batch_minkowski_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), p=2, dim=1)
        tensor([[0.0000, 2.8284],
                [2.8284, 0.0000]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')
    if p is None:
        raise ValueError('p must be provided and cannot be None...')
    if isinstance(p, str) and p not in ['fro', 'nuc']:
        raise ValueError('p must be either "fro" or "nuc" or torch.inf (numpy.inf) or -torch.inf (numpy.inf) or numbers...')

    if dim == 1:
        x = x.T
    b, m = x.shape

    if centered:
        x = x - torch.mean(x, dim=0, keepdim=True)
    x_expanded_1 = x.unsqueeze(2)
    x_expanded_2 = x.unsqueeze(1)
    distance_matrix = torch.norm(x_expanded_1 - x_expanded_2, p=p, dim=0)

    assert distance_matrix.shape == (m, m)
    return distance_matrix

Computes the Manhattan distance kernel, which is the exponential of the negative Manhattan distance.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed when in batch mode.

 0

Returns:

Type Description
Tensor

The Manhattan distance kernel. Shape depends on whether x2 is provided or not: - If x2 is provided: A scalar kernel value. - If x2 is None: A kernel matrix of shape (m, m).

Examples:

>>> manhattan_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.0183)

>>> manhattan_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def manhattan_distance_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Manhattan distance kernel, which is the exponential of the negative Manhattan distance.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed when in batch mode.

        Returns
        -------
        torch.Tensor
            The Manhattan distance kernel. Shape depends on whether x2 is provided or not:
            - If x2 is provided: A scalar kernel value.
            - If x2 is None: A kernel matrix of shape (m, m).

        Examples
        --------
        >>> manhattan_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.0183)

        >>> manhattan_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    return torch.exp(-manhattan_distance(x=x, x2=x2, centered=centered, dim=dim))

Computes the Manhattan distance between tensors, either as a batch or for a single pair.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed when in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar distance value.
  • For batch computation (x2 is None): A distance matrix of shape (m, m).

Examples:

Pairwise Manhattan distance:

>>> manhattan_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(4.)

Batch Manhattan distance:

>>> manhattan_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0., 4.],
        [4., 0.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def manhattan_distance(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Manhattan distance between tensors, either as a batch or for a single pair.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed when in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar distance value.
            - For batch computation (x2 is None): A distance matrix of shape (m, m).

        Examples
        --------
        Pairwise Manhattan distance:
        >>> manhattan_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(4.)

        Batch Manhattan distance:
        >>> manhattan_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0., 4.],
                [4., 0.]])
    """
    if x2 is None:
        return batch_manhattan_distance(x=x, centered=centered, dim=dim)
    else:
        return instance_manhattan_distance(x1=x, x2=x2)

Computes the Manhattan distance (L1 norm) between two tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor. Must be 1-dimensional.

 required
x2 Tensor

The second input tensor. Must be 1-dimensional and have the same shape as x1.

 required

Returns:

Type Description
Tensor

The Manhattan distance between x1 and x2.

Examples:

>>> instance_manhattan_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(4.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_manhattan_distance(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the Manhattan distance (L1 norm) between two tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor. Must be 1-dimensional.
        x2 : torch.Tensor
            The second input tensor. Must be 1-dimensional and have the same shape as x1.

        Returns
        -------
        torch.Tensor
            The Manhattan distance between x1 and x2.

        Examples
        --------
        >>> instance_manhattan_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(4.)
    """
    return instance_minkowski_distance(x1=x1, x2=x2, p=1)

Computes the Manhattan distance (L1 norm) between all pairs of rows in a batch of tensors.

Parameters:

Name Type Description Default
x Tensor

The input tensor. Must be 2-dimensional.

 required
centered bool

Whether to center the data when computing distances.

 False
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A matrix of pairwise Manhattan distances with shape (m, m), where m is the number of rows in x.

Examples:

>>> batch_manhattan_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0., 4.],
        [4., 0.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_manhattan_distance(x: torch.Tensor, centered: bool = False, dim: int = 0):
    """
        Computes the Manhattan distance (L1 norm) between all pairs of rows in a batch of tensors.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. Must be 2-dimensional.
        centered : bool, default=False
            Whether to center the data when computing distances.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A matrix of pairwise Manhattan distances with shape (m, m), where m is the number of rows in x.

        Examples
        --------
        >>> batch_manhattan_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0., 4.],
                [4., 0.]])
    """
    return batch_minkowski_distance(x=x, p=1, centered=centered, dim=dim)

Computes the Euclidean distance kernel, which is the exponential of the negative Euclidean distance.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor

The Euclidean distance kernel. Shape depends on whether x2 is provided: - If x2 is provided: A scalar kernel value. - If x2 is None: A kernel matrix of shape (m, m).

Examples:

>>> euclidean_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.0591)

>>> euclidean_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.0591],
        [0.0591, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def euclidean_distance_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Euclidean distance kernel, which is the exponential of the negative Euclidean distance.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            The Euclidean distance kernel. Shape depends on whether x2 is provided:
            - If x2 is provided: A scalar kernel value.
            - If x2 is None: A kernel matrix of shape (m, m).

        Examples
        --------
        >>> euclidean_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.0591)

        >>> euclidean_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.0591],
                [0.0591, 1.0000]])
    """
    return torch.exp(-euclidean_distance(x=x, x2=x2, centered=centered, dim=dim))

Computes the Euclidean distance between tensors, either as a batch or for a single pair.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar distance value.
  • For batch computation (x2 is None): A distance matrix of shape (m, m).

Examples:

Pairwise Euclidean distance:

>>> euclidean_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(2.8284)

Batch Euclidean distance:

>>> euclidean_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0.0000, 2.8284],
        [2.8284, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def euclidean_distance(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Euclidean distance between tensors, either as a batch or for a single pair.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar distance value.
            - For batch computation (x2 is None): A distance matrix of shape (m, m).

        Examples
        --------
        Pairwise Euclidean distance:
        >>> euclidean_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(2.8284)

        Batch Euclidean distance:
        >>> euclidean_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0.0000, 2.8284],
                [2.8284, 0.0000]])
    """
    if x2 is None:
        return batch_euclidean_distance(x=x, centered=centered, dim=dim)
    else:
        return instance_euclidean_distance(x1=x, x2=x2)

Computes the Euclidean distance (L2 norm) between two tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor. Must be 1-dimensional.

 required
x2 Tensor

The second input tensor. Must be 1-dimensional and have the same shape as x1.

 required

Returns:

Type Description
Tensor

The Euclidean distance between x1 and x2.

Examples:

>>> instance_euclidean_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(2.8284)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_euclidean_distance(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the Euclidean distance (L2 norm) between two tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor. Must be 1-dimensional.
        x2 : torch.Tensor
            The second input tensor. Must be 1-dimensional and have the same shape as x1.

        Returns
        -------
        torch.Tensor
            The Euclidean distance between x1 and x2.

        Examples
        --------
        >>> instance_euclidean_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(2.8284)
    """
    return instance_minkowski_distance(x1=x1, x2=x2, p=2)

Computes the Euclidean distance (L2 norm) between all pairs of rows in a batch of tensors.

Parameters:

Name Type Description Default
x Tensor

The input tensor. Must be 2-dimensional.

 required
centered bool

Whether to center the data when computing distances.

 False
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A matrix of pairwise Euclidean distances with shape (m, m), where m is the number of rows in x.

Examples:

>>> batch_euclidean_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0.0000, 2.8284],
        [2.8284, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_euclidean_distance(x: torch.Tensor, centered: bool = False, dim: int = 0):
    """
        Computes the Euclidean distance (L2 norm) between all pairs of rows in a batch of tensors.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. Must be 2-dimensional.
        centered : bool, default=False
            Whether to center the data when computing distances.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A matrix of pairwise Euclidean distances with shape (m, m), where m is the number of rows in x.

        Examples
        --------
        >>> batch_euclidean_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0.0000, 2.8284],
                [2.8284, 0.0000]])
    """
    return batch_minkowski_distance(x=x, p=2, centered=centered, dim=dim)

Computes the Chebyshev distance kernel, which is the exponential of the negative Chebyshev distance.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor

The Chebyshev distance kernel. Shape depends on whether x2 is provided: - If x2 is provided: A scalar kernel value. - If x2 is None: A kernel matrix of shape (m, m).

Examples:

Pairwise Chebyshev distance kernel:

>>> chebyshev_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.1353)

Batch Chebyshev distance kernel:

>>> chebyshev_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.1353],
        [0.1353, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def chebyshev_distance_kernel(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Chebyshev distance kernel, which is the exponential of the negative Chebyshev distance.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            The Chebyshev distance kernel. Shape depends on whether x2 is provided:
            - If x2 is provided: A scalar kernel value.
            - If x2 is None: A kernel matrix of shape (m, m).

        Examples
        --------
        Pairwise Chebyshev distance kernel:
        >>> chebyshev_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.1353)

        Batch Chebyshev distance kernel:
        >>> chebyshev_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.1353],
                [0.1353, 1.0000]])
    """
    return torch.exp(-chebyshev_distance(x=x, x2=x2, centered=centered, dim=dim))

Computes the Chebyshev distance between tensors, either as a batch or for a single pair.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
centered bool

Whether to center the data when computing distances in batch mode.

 False
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar distance value.
  • For batch computation (x2 is None): A distance matrix of shape (m, m).

Examples:

Pairwise Chebyshev distance:

>>> chebyshev_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(2.)

Batch Chebyshev distance:

>>> chebyshev_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0., 2.],
        [2., 0.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def chebyshev_distance(x: torch.Tensor, x2: torch.Tensor = None, centered: bool = False, dim: int = 0):
    """
        Computes the Chebyshev distance between tensors, either as a batch or for a single pair.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        centered : bool, default=False
            Whether to center the data when computing distances in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar distance value.
            - For batch computation (x2 is None): A distance matrix of shape (m, m).

        Examples
        --------
        Pairwise Chebyshev distance:
        >>> chebyshev_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(2.)

        Batch Chebyshev distance:
        >>> chebyshev_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0., 2.],
                [2., 0.]])
    """
    if x2 is None:
        return batch_chebyshev_distance(x=x, centered=centered, dim=dim)
    else:
        return instance_chebyshev_distance(x1=x, x2=x2)

Computes the Chebyshev distance between two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required

Returns:

Type Description
Tensor

The Chebyshev distance as a scalar value.

Examples:

>>> instance_chebyshev_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 5.0]))
tensor(3.)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_chebyshev_distance(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the Chebyshev distance between two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.

        Returns
        -------
        torch.Tensor
            The Chebyshev distance as a scalar value.

        Examples
        --------
        >>> instance_chebyshev_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 5.0]))
        tensor(3.)
    """
    return instance_minkowski_distance(x1=x1, x2=x2, p=torch.inf)

Computes the Chebyshev distance matrix for a batch of vectors.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
centered bool

Whether to center the data.

 False
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A distance matrix of shape (m, m).

Examples:

>>> batch_chebyshev_distance(torch.tensor([[1.0, 2.0], [3.0, 5.0]]), dim=1)
tensor([[0., 3.],
        [3., 0.]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_chebyshev_distance(x: torch.Tensor, centered: bool = False, dim: int = 0):
    """
        Computes the Chebyshev distance matrix for a batch of vectors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        centered : bool, default=False
            Whether to center the data.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A distance matrix of shape (m, m).

        Examples
        --------
        >>> batch_chebyshev_distance(torch.tensor([[1.0, 2.0], [3.0, 5.0]]), dim=1)
        tensor([[0., 3.],
                [3., 0.]])
    """
    return batch_minkowski_distance(x=x, p=torch.inf, centered=centered, dim=dim)

Computes the Canberra distance kernel, which is the exponential of the negative Canberra distance.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor

The Canberra distance kernel. Shape depends on whether x2 is provided: - If x2 is provided: A scalar kernel value. - If x2 is None: A kernel matrix of shape (m, m).

Examples:

Pairwise Canberra distance kernel:

>>> canberra_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.4346)

Batch Canberra distance kernel:

>>> canberra_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[1.0000, 0.4346],
        [0.4346, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def canberra_distance_kernel(x: torch.Tensor, x2: torch.Tensor = None, dim: int = 0):
    """
        Computes the Canberra distance kernel, which is the exponential of the negative Canberra distance.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            The Canberra distance kernel. Shape depends on whether x2 is provided:
            - If x2 is provided: A scalar kernel value.
            - If x2 is None: A kernel matrix of shape (m, m).

        Examples
        --------
        Pairwise Canberra distance kernel:
        >>> canberra_distance_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.4346)

        Batch Canberra distance kernel:
        >>> canberra_distance_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[1.0000, 0.4346],
                [0.4346, 1.0000]])
    """
    return torch.exp(-canberra_distance(x=x, x2=x2, dim=dim))

Computes the Canberra distance between tensors, either as a batch or for a single pair.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar distance value.
  • For batch computation (x2 is None): A distance matrix of shape (m, m).

Examples:

Pairwise Canberra distance:

>>> canberra_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.8333)

Batch Canberra distance:

>>> canberra_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0.0000, 0.8333],
        [0.8333, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def canberra_distance(x: torch.Tensor, x2: torch.Tensor = None, dim: int = 0):
    """
        Computes the Canberra distance between tensors, either as a batch or for a single pair.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar distance value.
            - For batch computation (x2 is None): A distance matrix of shape (m, m).

        Examples
        --------
        Pairwise Canberra distance:
        >>> canberra_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.8333)

        Batch Canberra distance:
        >>> canberra_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0.0000, 0.8333],
                [0.8333, 0.0000]])
    """
    if x2 is None:
        return batch_canberra_distance(x=x, dim=dim)
    else:
        return instance_canberra_distance(x1=x, x2=x2)

Computes the Canberra distance between two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required

Returns:

Type Description
Tensor

The Canberra distance as a scalar value.

Examples:

>>> instance_canberra_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
tensor(0.8333)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_canberra_distance(x1: torch.Tensor, x2: torch.Tensor, *args, **kwargs):
    """
        Computes the Canberra distance between two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.

        Returns
        -------
        torch.Tensor
            The Canberra distance as a scalar value.

        Examples
        --------
        >>> instance_canberra_distance(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]))
        tensor(0.8333)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')

    numerator = torch.absolute(x1 - x2)
    denominator = torch.absolute(x1) + torch.absolute(x2)
    canberra_dist = torch.sum(numerator / (denominator + 1e-10))

    return canberra_dist

Computes the Canberra distance matrix for a batch of vectors.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A distance matrix of shape (m, m).

Examples:

>>> batch_canberra_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
tensor([[0.0000, 0.8333],
        [0.8333, 0.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_canberra_distance(x: torch.Tensor, dim: int = 0):
    """
        Computes the Canberra distance matrix for a batch of vectors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A distance matrix of shape (m, m).

        Examples
        --------
        >>> batch_canberra_distance(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), dim=1)
        tensor([[0.0000, 0.8333],
                [0.8333, 0.0000]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if dim == 1:
        x = x.T
    b, m = x.shape

    x_expanded_1 = torch.unsqueeze(x, dim=2)
    x_expanded_2 = torch.unsqueeze(x, dim=1)

    numerator = torch.absolute(x_expanded_1 - x_expanded_2)
    denominator = torch.absolute(x_expanded_1) + torch.absolute(x_expanded_2)
    canberra_dist_matrix = torch.sum(numerator / (denominator + 1e-10), dim=0)

    assert canberra_dist_matrix.shape == (m, m)
    return canberra_dist_matrix

Computes the exponential kernel, which is an RBF-like kernel based on the exponential of squared Euclidean distance.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.

 None
gamma float

The scale parameter for the kernel.

 1.0
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar kernel value.
  • For batch computation (x2 is None): A kernel matrix of shape (m, m).

Examples:

Pairwise exponential kernel:

>>> exponential_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), gamma=0.5)
tensor(0.0183)

Batch exponential kernel:

>>> exponential_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), gamma=0.5, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def exponential_kernel(x: torch.Tensor, x2: torch.Tensor = None, gamma: float = 1.0, dim: int = 0):
    """
        Computes the exponential kernel, which is an RBF-like kernel based on the exponential of squared Euclidean distance.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, distances are computed in batch mode.
        gamma : float, default=1.0
            The scale parameter for the kernel.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar kernel value.
            - For batch computation (x2 is None): A kernel matrix of shape (m, m).

        Examples
        --------
        Pairwise exponential kernel:
        >>> exponential_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), gamma=0.5)
        tensor(0.0183)

        Batch exponential kernel:
        >>> exponential_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), gamma=0.5, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if x2 is None:
        return batch_exponential_kernel(x=x, gamma=gamma, dim=dim)
    else:
        return instance_exponential_kernel(x1=x, x2=x2, gamma=gamma)

Computes the exponential kernel for a single pair of tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
gamma float

The scale parameter for the kernel.

 1.0

Returns:

Type Description
Tensor

The kernel value as a scalar tensor.

Examples:

>>> instance_exponential_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), gamma=0.5)
tensor(0.0183)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_exponential_kernel(x1: torch.Tensor, x2: torch.Tensor, gamma: float = 1.0):
    """
        Computes the exponential kernel for a single pair of tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        gamma : float, default=1.0
            The scale parameter for the kernel.

        Returns
        -------
        torch.Tensor
            The kernel value as a scalar tensor.

        Examples
        --------
        >>> instance_exponential_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), gamma=0.5)
        tensor(0.0183)
    """
    return torch.exp(-gamma*instance_euclidean_distance(x1=x1, x2=x2)**2)

Computes the exponential kernel in batch mode for a set of tensors.

Parameters:

Name Type Description Default
x Tensor

The input tensor. Must be a 2D tensor with shape (b, m).

 required
gamma float

The scale parameter for the kernel.

 1.0
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A kernel matrix of shape (m, m).

Examples:

>>> batch_exponential_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), gamma=0.5, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_exponential_kernel(x: torch.Tensor, gamma: float = 1.0, dim: int = 0):
    """
        Computes the exponential kernel in batch mode for a set of tensors.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. Must be a 2D tensor with shape (b, m).
        gamma : float, default=1.0
            The scale parameter for the kernel.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A kernel matrix of shape (m, m).

        Examples
        --------
        >>> batch_exponential_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), gamma=0.5, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    return torch.exp(-gamma*batch_euclidean_distance(x=x, dim=dim)**2)

Computes the Gaussian Radial Basis Function (RBF) kernel.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.

 None
sigma float

The standard deviation parameter for the RBF kernel.

 1.0
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar kernel value.
  • For batch computation (x2 is None): A kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If sigma is less than or equal to zero.

Examples:

Pairwise Gaussian RBF kernel:

>>> gaussian_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
tensor(0.0183)

Batch Gaussian RBF kernel:

>>> gaussian_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def gaussian_rbf_kernel(x: torch.Tensor, x2: torch.Tensor = None, sigma: float = 1.0, dim: int = 0):
    """
        Computes the Gaussian Radial Basis Function (RBF) kernel.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.
        sigma : float, default=1.0
            The standard deviation parameter for the RBF kernel.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar kernel value.
            - For batch computation (x2 is None): A kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If `sigma` is less than or equal to zero.

        Examples
        --------
        Pairwise Gaussian RBF kernel:
        >>> gaussian_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
        tensor(0.0183)

        Batch Gaussian RBF kernel:
        >>> gaussian_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if x2 is None:
        return batch_gaussian_rbf_kernel(x=x, sigma=sigma, dim=dim)
    else:
        return instance_gaussian_rbf_kernel(x1=x, x2=x2, sigma=sigma)

Computes the Gaussian Radial Basis Function (RBF) kernel for two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
sigma float

The standard deviation of the Gaussian kernel.

 1.0

Returns:

Type Description
Tensor

The Gaussian RBF kernel value.

Examples:

>>> instance_gaussian_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
tensor(0.0183)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_gaussian_rbf_kernel(x1: torch.Tensor, x2: torch.Tensor, sigma: float = 1.0):
    """
        Computes the Gaussian Radial Basis Function (RBF) kernel for two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        sigma : float, default=1.0
            The standard deviation of the Gaussian kernel.

        Returns
        -------
        torch.Tensor
            The Gaussian RBF kernel value.

        Examples
        --------
        >>> instance_gaussian_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
        tensor(0.0183)
    """
    if sigma <= 0.0:
        raise ValueError('sigma must be positive...')
    return torch.exp(- instance_euclidean_distance(x1=x1, x2=x2)**2 / (2 * sigma**2))

Computes the Gaussian Radial Basis Function (RBF) kernel matrix for a batch of vectors.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
sigma float

The standard deviation of the Gaussian kernel.

 1.0
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The Gaussian RBF kernel matrix of shape (m, m).

Examples:

>>> batch_gaussian_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_gaussian_rbf_kernel(x: torch.Tensor, sigma: float = 1.0, dim: int = 0):
    """
        Computes the Gaussian Radial Basis Function (RBF) kernel matrix for a batch of vectors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        sigma : float, default=1.0
            The standard deviation of the Gaussian kernel.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The Gaussian RBF kernel matrix of shape (m, m).

        Examples
        --------
        >>> batch_gaussian_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if sigma <= 0.0:
        raise ValueError('sigma must be positive...')
    return torch.exp(- batch_euclidean_distance(x=x, dim=dim) ** 2 / (2 * sigma ** 2))

Computes the Laplacian kernel.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.

 None
sigma float

The scale parameter for the Laplacian kernel.

 1.0
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar kernel value.
  • For batch computation (x2 is None): A kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If sigma is zero.

Examples:

Pairwise Laplacian kernel:

>>> laplacian_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
tensor(0.0183)

Batch Laplacian kernel:

>>> laplacian_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def laplacian_kernel(x: torch.Tensor, x2: torch.Tensor = None, sigma: float = 1.0, dim: int = 0):
    """
        Computes the Laplacian kernel.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.
        sigma : float, default=1.0
            The scale parameter for the Laplacian kernel.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar kernel value.
            - For batch computation (x2 is None): A kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If `sigma` is zero.

        Examples
        --------
        Pairwise Laplacian kernel:
        >>> laplacian_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
        tensor(0.0183)

        Batch Laplacian kernel:
        >>> laplacian_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if x2 is None:
        return batch_laplacian_kernel(x=x, sigma=sigma, dim=dim)
    else:
        return instance_laplacian_kernel(x1=x, x2=x2, sigma=sigma)

Computes the Laplacian kernel for two 1D tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor.

 required
x2 Tensor

The second input tensor.

 required
sigma float

The kernel bandwidth parameter.

 1.0

Returns:

Type Description
Tensor

The Laplacian kernel value.

Examples:

>>> instance_laplacian_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
tensor(0.0183)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_laplacian_kernel(x1: torch.Tensor, x2: torch.Tensor, sigma: float = 1.0):
    """
        Computes the Laplacian kernel for two 1D tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor.
        x2 : torch.Tensor
            The second input tensor.
        sigma : float, default=1.0
            The kernel bandwidth parameter.

        Returns
        -------
        torch.Tensor
            The Laplacian kernel value.

        Examples
        --------
        >>> instance_laplacian_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), sigma=1.0)
        tensor(0.0183)
    """
    if sigma == 0:
        raise ValueError('sigma must be not be zero...')
    return torch.exp(- instance_manhattan_distance(x1=x1, x2=x2)/sigma)

Computes the Laplacian kernel matrix for a batch of vectors.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (b, m).

 required
sigma float

The kernel bandwidth parameter.

 1.0
dim int

The dimension along which the kernel is computed.

 0

Returns:

Type Description
Tensor

The Laplacian kernel matrix of shape (m, m).

Examples:

>>> batch_laplacian_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_laplacian_kernel(x: torch.Tensor, sigma: float = 1.0, dim: int = 0):
    """
        Computes the Laplacian kernel matrix for a batch of vectors.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape (b, m).
        sigma : float, default=1.0
            The kernel bandwidth parameter.
        dim : int, default=0
            The dimension along which the kernel is computed.

        Returns
        -------
        torch.Tensor
            The Laplacian kernel matrix of shape (m, m).

        Examples
        --------
        >>> batch_laplacian_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), sigma=1.0, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if sigma == 0:
        raise ValueError('sigma must be not be zero...')
    return torch.exp(- batch_manhattan_distance(x=x, dim=dim)/sigma)

Computes the anisotropic radial basis function (RBF) kernel.

Parameters:

Name Type Description Default
x Tensor

The input tensor. For pairwise computation, this is the first tensor (x1).

 required
x2 Tensor

The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.

 None
a_vector Tensor

A vector defining the anisotropy in each dimension. Must have the same shape as x.

 None
a_scalar float

A scalar defining the uniform anisotropy. Ignored if a_vector is provided.

 1.0
dim int

The dimension along which the distances are computed in batch mode.

 0

Returns:

Type Description
Tensor
  • For pairwise computation (x and x2 provided): A scalar kernel value.
  • For batch computation (x2 is None): A kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If input tensors are empty or a_vector is incompatible with x.
Warnings

If a_vector is all zeros or a_scalar is zero.

Examples:

Pairwise anisotropic RBF kernel:

>>> anisotropic_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), a_vector=torch.tensor([0.5, 0.5]))
tensor(0.0183)

Batch anisotropic RBF kernel:

>>> anisotropic_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), a_scalar=0.5, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def anisotropic_rbf_kernel(x: torch.Tensor, x2: torch.Tensor = None, a_vector: torch.Tensor = None, a_scalar: float = 1.0, dim: int = 0):
    """
        Computes the anisotropic radial basis function (RBF) kernel.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor. For pairwise computation, this is the first tensor (x1).
        x2 : torch.Tensor, optional
            The second input tensor for pairwise computation. If not provided, the kernel is computed in batch mode.
        a_vector : torch.Tensor, optional
            A vector defining the anisotropy in each dimension. Must have the same shape as `x`.
        a_scalar : float, default=1.0
            A scalar defining the uniform anisotropy. Ignored if `a_vector` is provided.
        dim : int, default=0
            The dimension along which the distances are computed in batch mode.

        Returns
        -------
        torch.Tensor
            - For pairwise computation (x and x2 provided): A scalar kernel value.
            - For batch computation (x2 is None): A kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If input tensors are empty or `a_vector` is incompatible with `x`.
        Warnings
            If `a_vector` is all zeros or `a_scalar` is zero.

        Examples
        --------
        Pairwise anisotropic RBF kernel:
        >>> anisotropic_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), a_vector=torch.tensor([0.5, 0.5]))
        tensor(0.0183)

        Batch anisotropic RBF kernel:
        >>> anisotropic_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), a_scalar=0.5, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if x2 is None:
        return batch_anisotropic_rbf_kernel(x=x, a_vector=a_vector, a_scalar=a_scalar, dim=dim)
    else:
        return instance_anisotropic_rbf_kernel(x1=x, x2=x2, a_vector=a_vector, a_scalar=a_scalar)

Computes the pairwise anisotropic RBF kernel for two input tensors.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor (1D).

 required
x2 Tensor

The second input tensor (1D).

 required
a_vector Tensor

A vector defining the anisotropy in each dimension. Must have the same shape as x1.

 None
a_scalar float

A scalar defining the uniform anisotropy. Ignored if a_vector is provided.

 1.0

Returns:

Type Description
Tensor

A scalar value of the kernel.

Raises:

Type Description
ValueError

If input tensors are empty or dimensions do not match.
Warnings

If a_vector is all zeros or a_scalar is zero.

Examples:

>>> instance_anisotropic_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), a_vector=torch.tensor([0.5, 0.5]))
tensor(0.0183)
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_anisotropic_rbf_kernel(x1: torch.Tensor, x2: torch.Tensor, a_vector: torch.Tensor = None, a_scalar: float = 1.0):
    """
        Computes the pairwise anisotropic RBF kernel for two input tensors.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor (1D).
        x2 : torch.Tensor
            The second input tensor (1D).
        a_vector : torch.Tensor, optional
            A vector defining the anisotropy in each dimension. Must have the same shape as `x1`.
        a_scalar : float, default=1.0
            A scalar defining the uniform anisotropy. Ignored if `a_vector` is provided.

        Returns
        -------
        torch.Tensor
            A scalar value of the kernel.

        Raises
        ------
        ValueError
            If input tensors are empty or dimensions do not match.
        Warnings
            If `a_vector` is all zeros or `a_scalar` is zero.

        Examples
        --------
        >>> instance_anisotropic_rbf_kernel(torch.tensor([1.0, 2.0]), torch.tensor([3.0, 4.0]), a_vector=torch.tensor([0.5, 0.5]))
        tensor(0.0183)
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')
    if a_vector is not None and a_vector.shape != x1.shape:
        raise ValueError('a_vector must be of dimension 1 and has identical shapes as x1 and x2...')
    if a_vector is not None and a_vector.numel() == 0:
        raise ValueError("Input a vector must not be None or empty...")

    if a_vector is not None and torch.all(a_vector == 0):
        warnings.warn('input a vector should not be all zeros...')
    if a_scalar is not None and a_scalar == 0.0:
        warnings.warn("Input a scalar must not be zero...")

    if a_vector is not None:
        a = a_vector
    else:
        a = a_scalar * torch.ones_like(x1)

    assert a.shape == x1.shape

    A = torch.diag(a)
    d = x1 - x2
    return torch.exp(-d @ A @ d.permute(*torch.arange(d.ndim - 1, -1, -1)))

Computes the batch anisotropic RBF kernel.

Parameters:

Name Type Description Default
x Tensor

The input tensor (2D).

 required
a_vector Tensor

A vector defining the anisotropy in each dimension. Must match the batch size of x.

 None
a_scalar float

A scalar defining the uniform anisotropy. Ignored if a_vector is provided.

 1.0
dim int

The dimension along which the distances are computed.

 0

Returns:

Type Description
Tensor

A kernel matrix of shape (m, m).

Raises:

Type Description
ValueError

If input tensors are empty or incompatible.
Warnings

If a_vector is all zeros or a_scalar is zero.

Examples:

>>> batch_anisotropic_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), a_scalar=0.5, dim=1)
tensor([[1.0000, 0.0183],
        [0.0183, 1.0000]])
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_anisotropic_rbf_kernel(x: torch.Tensor, a_vector: torch.Tensor = None, a_scalar: float = 1.0, dim: int = 0):
    """
        Computes the batch anisotropic RBF kernel.

        Parameters
        ----------
        x : torch.Tensor
            The input tensor (2D).
        a_vector : torch.Tensor, optional
            A vector defining the anisotropy in each dimension. Must match the batch size of `x`.
        a_scalar : float, default=1.0
            A scalar defining the uniform anisotropy. Ignored if `a_vector` is provided.
        dim : int, default=0
            The dimension along which the distances are computed.

        Returns
        -------
        torch.Tensor
            A kernel matrix of shape (m, m).

        Raises
        ------
        ValueError
            If input tensors are empty or incompatible.
        Warnings
            If `a_vector` is all zeros or `a_scalar` is zero.

        Examples
        --------
        >>> batch_anisotropic_rbf_kernel(torch.tensor([[1.0, 2.0], [3.0, 4.0]]), a_scalar=0.5, dim=1)
        tensor([[1.0000, 0.0183],
                [0.0183, 1.0000]])
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')
    if a_vector is not None and a_vector.numel() == 0:
        raise ValueError("Input a vector must not be None or empty...")

    if a_vector is not None and torch.all(a_vector == 0):
        warnings.warn('input a vector should not be all zeros...')
    if a_scalar is not None and a_scalar == 0.0:
        warnings.warn("Input a scalar must not be zero...")

    if dim == 1:
        x = x.T
    b, m = x.shape

    if a_vector is not None:
        a = a_vector
    else:
        a = a_scalar * torch.ones(b)

    assert a is not None and a.ndim == 1 and a.size(0) == b

    A = torch.diag(a)

    x = x.T  # Shape: (m, b)

    diff_matrix = x[:, None, :] - x[None, :, :]  # Shape: (m, m, b)
    rbf_matrix = torch.einsum('ijk,kl,ijl->ij', diff_matrix, A, diff_matrix)

    assert rbf_matrix.shape == (m, m)
    return torch.exp(-rbf_matrix)

Computes a custom hybrid kernel by combining multiple kernel functions with specified weights.

Parameters:

Name Type Description Default
x Tensor

The first input tensor. Can be a batch or single instance depending on x2.

 required
x2 Tensor

The second input tensor. If None, the kernel will be computed in batch mode.

 None
kernels List[Callable[[Tensor, Tensor], Tensor]]

A list of kernel functions to be combined.

 None
weights Union[List, Tuple, float, Parameter]

Weights for combining the kernel functions. If None, all kernels are equally weighted.

 None
dim int

Dimension to apply the kernel operation. Default is 0.

 0

Returns:

Type Description
Tensor

The computed hybrid kernel output.

Raises:

Type Description
ValueError

If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
Source code in tinybig/koala/linear_algebra/kernel.py 
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def custom_hybrid_kernel(x: torch.Tensor, x2: torch.Tensor = None, kernels: List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]] = None, weights: Union[List, Tuple, float, torch.nn.Parameter] = None, dim: int = 0):
    """
        Computes a custom hybrid kernel by combining multiple kernel functions with specified weights.

        Parameters
        ----------
        x : torch.Tensor
            The first input tensor. Can be a batch or single instance depending on `x2`.
        x2 : torch.Tensor, optional
            The second input tensor. If `None`, the kernel will be computed in batch mode.
        kernels : List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]]
            A list of kernel functions to be combined.
        weights : Union[List, Tuple, float, torch.nn.Parameter], optional
            Weights for combining the kernel functions. If `None`, all kernels are equally weighted.
        dim : int, optional
            Dimension to apply the kernel operation. Default is `0`.

        Returns
        -------
        torch.Tensor
            The computed hybrid kernel output.

        Raises
        ------
        ValueError
            If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
    """
    if x2 is None:
        return batch_custom_hybrid_kernel(x=x, kernels=kernels, weights=weights, dim=dim)
    else:
        return instance_custom_hybrid_kernel(x1=x, x2=x2, kernels=kernels, weights=weights)

Computes a hybrid kernel for two single instances using a combination of kernel functions.

Parameters:

Name Type Description Default
x1 Tensor

The first input tensor (single instance).

 required
x2 Tensor

The second input tensor (single instance).

 required
kernels List[Callable[[Tensor, Tensor], Tensor]]

A list of kernel functions to be combined.

 required
weights Union[List, Tuple, float]

Weights for combining the kernel functions. If None, all kernels are equally weighted.

 None

Returns:

Type Description
Tensor

The computed hybrid kernel value for the two instances.

Raises:

Type Description
ValueError

If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
Source code in tinybig/koala/linear_algebra/kernel.py 
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def instance_custom_hybrid_kernel(x1: torch.Tensor, x2: torch.Tensor, kernels: List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]], weights: Union[List, Tuple, float] = None):
    """
        Computes a hybrid kernel for two single instances using a combination of kernel functions.

        Parameters
        ----------
        x1 : torch.Tensor
            The first input tensor (single instance).
        x2 : torch.Tensor
            The second input tensor (single instance).
        kernels : List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]]
            A list of kernel functions to be combined.
        weights : Union[List, Tuple, float], optional
            Weights for combining the kernel functions. If `None`, all kernels are equally weighted.

        Returns
        -------
        torch.Tensor
            The computed hybrid kernel value for the two instances.

        Raises
        ------
        ValueError
            If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
    """
    if x1 is None or x2 is None or x1.numel() == 0 or x2.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x1.ndim != 1 or x1.shape != x2.shape:
        raise ValueError('x1 and x2 must be of dimension 1...')

    if kernels is None or len(kernels) == 0:
        raise ValueError("At least one kernel function must be provided.")
    elif not isinstance(kernels, list):
        kernels = [kernels]

    if weights is None:
        weights = [1 / len(kernels)] * len(kernels)
    elif not isinstance(weights, list):
        weights = [weights]

    if len(kernels) != len(weights):
        raise ValueError("The number of kernels must match the number of weights.")

    kernel_outputs = [kernel(x1, x2) for kernel in kernels]
    shapes = [output.shape for output in kernel_outputs]
    if len(set(shapes)) != 1:
        raise ValueError("All kernel outputs must have the same shape.")

    weighted_sum = sum(weight * output for weight, output in zip(weights, kernel_outputs))
    return weighted_sum

Computes a hybrid kernel for a batch of instances using a combination of kernel functions.

Parameters:

Name Type Description Default
x Tensor

Input tensor of shape (batch_size, feature_dim) for batch processing.

 required
kernels List[Callable[[Tensor, Tensor], Tensor]]

A list of kernel functions to be combined.

 required
weights Union[List, Tuple, float]

Weights for combining the kernel functions. If None, all kernels are equally weighted.

 None
dim int

Dimension to apply the kernel operation. Default is 0.

 0

Returns:

Type Description
Tensor

The computed hybrid kernel matrix for the batch.

Raises:

Type Description
ValueError

If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
Source code in tinybig/koala/linear_algebra/kernel.py 
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def batch_custom_hybrid_kernel(x: torch.Tensor, kernels: List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]], weights: Union[List, Tuple, float] = None, dim: int = 0):
    """
        Computes a hybrid kernel for a batch of instances using a combination of kernel functions.

        Parameters
        ----------
        x : torch.Tensor
            Input tensor of shape `(batch_size, feature_dim)` for batch processing.
        kernels : List[Callable[[torch.Tensor, torch.Tensor], torch.Tensor]]
            A list of kernel functions to be combined.
        weights : Union[List, Tuple, float], optional
            Weights for combining the kernel functions. If `None`, all kernels are equally weighted.
        dim : int, optional
            Dimension to apply the kernel operation. Default is `0`.

        Returns
        -------
        torch.Tensor
            The computed hybrid kernel matrix for the batch.

        Raises
        ------
        ValueError
            If input tensors are invalid, no kernels are provided, or weights and kernels mismatch.
    """
    if x is None or x.numel() == 0:
        raise ValueError("Input tensors must not be None or empty...")
    if x.ndim != 2:
        raise ValueError('x must be of dimension 2...')
    if dim not in [0, 1]:
        raise ValueError('dim must be 0 or 1')

    if kernels is None or len(kernels) == 0:
        raise ValueError("At least one kernel function must be provided.")
    elif not isinstance(kernels, list):
        kernels = [kernels]

    if weights is None:
        weights = [1 / len(kernels)] * len(kernels)
    elif not isinstance(weights, list):
        weights = [weights]

    if len(kernels) != len(weights):
        raise ValueError("The number of kernels must match the number of weights.")

    kernel_outputs = [kernel(x, dim=dim) for kernel in kernels]

    shapes = [output.shape for output in kernel_outputs]
    if len(set(shapes)) != 1:
        raise ValueError("All kernel outputs must have the same shape.")

    weighted_sum = sum(weight * output for weight, output in zip(weights, kernel_outputs))
    return weighted_sum