random_matrix_adaption_parameterized_concatenation_fusion

Bases: parameterized_concatenation_fusion

A parameterized concatenation fusion with random matrix adaptation for parameter fabrication.

Notes

Formally, given input interdependence matrices \(\mathbf{A}_1, \mathbf{A}_2, \ldots, \mathbf{A}_k\), where each matrix \(\mathbf{A}_i \in R^{m \times n_i}\) has \(m\) rows and \(n_i\) columns, we define the fusion operator as follows:

\[
    \begin{equation}
    \begin{aligned}
    \mathbf{A} &= \text{fusion}(\mathbf{A}_1, \mathbf{A}_2, \cdots, \mathbf{A}_k) \\
    &= \left( \mathbf{A}_1 \sqcup \mathbf{A}_2 \sqcup \cdots \sqcup \mathbf{A}_k \right) \mathbf{W} \in R^{m \times n},
    \end{aligned}
    \end{equation}
\]

where \(\sqcup\) denotes the row-wise concatenation of the matrices.

Notation \(\mathbf{W} \in R^{(\sum_{i=1}^k n_i) \times n}\) denotes the parameter matrix fabricated from the learnable parameter vector \(\mathbf{w} \in R^{l}\), which can be represented as follows:

\[
    \begin{equation}
    \psi(\mathbf{w}) = \Lambda_1 \mathbf{A} \Lambda_1 \mathbf{B}^\top \in R^{(\sum_{i=1}^k n_i) \times n},
    \end{equation}
\]

Notations \(\Lambda_1\) and \(\Lambda_2\) denote two diagonal matrices \(\Lambda_1 = diag( \lambda_1) \in R^{(\sum_{i=1}^k n_i) \times (\sum_{i=1}^k n_i)}\) and \(\Lambda_2 = diag(\lambda_2) \in R^{r \times r}\) where the diagonal vectors \(\lambda_1\) and \(\lambda_2\) are partitioned from the parameter vector \(\mathbf{w}\). Matrices \(\mathbf{A} \in R^{(\sum_{i=1}^k n_i) \times r}\) and \(\mathbf{B} \in R^{r \times n}\) are randomly sampled from the Gaussian distribution \(\mathcal{N}(\mathbf{0}, \mathbf{I})\).

The required length of parameter vector of this interdependence function is \(\mathbf{w}\) is \(l = (\sum_{i=1}^k n_i) + r\).

Attributes:

Name	Type	Description
r	int	Rank for the random matrix adaptation.

Methods:

Name	Description
__init__	Initializes the random matrix adaptation parameterized concatenation fusion function.

Source code in tinybig/fusion/parameterized_concatenation_fusion.py

class random_matrix_adaption_parameterized_concatenation_fusion(parameterized_concatenation_fusion):
    r"""
        A parameterized concatenation fusion with random matrix adaptation for parameter fabrication.

        Notes
        ----------

        Formally, given input interdependence matrices $\mathbf{A}_1, \mathbf{A}_2, \ldots, \mathbf{A}_k$,
        where each matrix $\mathbf{A}_i \in R^{m \times n_i}$ has $m$ rows and $n_i$ columns,
        we define the fusion operator as follows:

        $$
            \begin{equation}
            \begin{aligned}
            \mathbf{A} &= \text{fusion}(\mathbf{A}_1, \mathbf{A}_2, \cdots, \mathbf{A}_k) \\
            &= \left( \mathbf{A}_1 \sqcup \mathbf{A}_2 \sqcup \cdots \sqcup \mathbf{A}_k \right) \mathbf{W} \in R^{m \times n},
            \end{aligned}
            \end{equation}
        $$

        where $\sqcup$ denotes the row-wise concatenation of the matrices.

        Notation $\mathbf{W} \in R^{(\sum_{i=1}^k n_i) \times n}$ denotes the parameter matrix fabricated from the learnable parameter vector $\mathbf{w} \in R^{l}$,
        which can be represented as follows:

        $$
            \begin{equation}
            \psi(\mathbf{w}) = \Lambda_1 \mathbf{A} \Lambda_1 \mathbf{B}^\top \in R^{(\sum_{i=1}^k n_i) \times n},
            \end{equation}
        $$

        Notations $\Lambda_1$ and $\Lambda_2$ denote two diagonal matrices $\Lambda_1 = diag( \lambda_1) \in R^{(\sum_{i=1}^k n_i) \times (\sum_{i=1}^k n_i)}$ and $\Lambda_2 = diag(\lambda_2) \in R^{r \times r}$
        where the diagonal vectors $\lambda_1$ and $\lambda_2$ are partitioned from the parameter vector $\mathbf{w}$.
        Matrices $\mathbf{A} \in R^{(\sum_{i=1}^k n_i) \times r}$ and $\mathbf{B} \in R^{r \times n}$ are randomly sampled from the Gaussian distribution $\mathcal{N}(\mathbf{0}, \mathbf{I})$.

        The required length of parameter vector of this interdependence function is $\mathbf{w}$ is $l = (\sum_{i=1}^k n_i) + r$.



        Attributes
        ----------
        r : int
            Rank for the random matrix adaptation.

        Methods
        -------
        __init__(...)
            Initializes the random matrix adaptation parameterized concatenation fusion function.
    """
    def __init__(self, r: int = 2, name: str = 'random_matrix_adaption_parameterized_concatenation_fusion', *args, **kwargs):
        """
            Initializes the random matrix adaptation parameterized concatenation fusion function.

            Parameters
            ----------
            r : int, optional
                Rank for the random matrix adaptation. Defaults to 2.
            name : str, optional
                Name of the fusion function. Defaults to "random_matrix_adaption_parameterized_concatenation_fusion".
            *args : tuple
                Additional positional arguments for the parent class.
            **kwargs : dict
                Additional keyword arguments for the parent class.
        """
        super().__init__(name=name, *args, **kwargs)
        self.r = r
        self.parameter_fabrication = random_matrix_adaption_reconciliation(r=self.r)

__init__(r=2, name='random_matrix_adaption_parameterized_concatenation_fusion', *args, **kwargs)

Initializes the random matrix adaptation parameterized concatenation fusion function.

Parameters:

Name	Type	Description	Default
r	int	Rank for the random matrix adaptation. Defaults to 2.	2
name	str	Name of the fusion function. Defaults to "random_matrix_adaption_parameterized_concatenation_fusion".	'random_matrix_adaption_parameterized_concatenation_fusion'
*args	tuple	Additional positional arguments for the parent class.	()
**kwargs	dict	Additional keyword arguments for the parent class.	{}

Source code in tinybig/fusion/parameterized_concatenation_fusion.py

def __init__(self, r: int = 2, name: str = 'random_matrix_adaption_parameterized_concatenation_fusion', *args, **kwargs):
    """
        Initializes the random matrix adaptation parameterized concatenation fusion function.

        Parameters
        ----------
        r : int, optional
            Rank for the random matrix adaptation. Defaults to 2.
        name : str, optional
            Name of the fusion function. Defaults to "random_matrix_adaption_parameterized_concatenation_fusion".
        *args : tuple
            Additional positional arguments for the parent class.
        **kwargs : dict
            Additional keyword arguments for the parent class.
    """
    super().__init__(name=name, *args, **kwargs)
    self.r = r
    self.parameter_fabrication = random_matrix_adaption_reconciliation(r=self.r)