lowrank_parameterized_concatenation_fusion

Bases: parameterized_concatenation_fusion

A parameterized concatenation fusion with a low-rank approximation 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:

$$ \begin{equation} \psi(\mathbf{w}) = \mathbf{A} \mathbf{B}^\top = \mathbf{W} \in R^{(\sum_{i=1}^k n_i) \times n}, \end{equation} $$ where \(\mathbf{A} \in R^{(\sum_{i=1}^k n_i) \times r}\) and \(\mathbf{B} \in R^{n \times r}\) are partitioned and reshaped from the parameter vector \(\mathbf{w}\).

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

Attributes:

Name	Type	Description
r	int	Rank of the low-rank approximation.

Methods:

Name	Description
__init__	Initializes the low-rank parameterized concatenation fusion function.

Source code in tinybig/fusion/parameterized_concatenation_fusion.py

class lowrank_parameterized_concatenation_fusion(parameterized_concatenation_fusion):
    r"""
        A parameterized concatenation fusion with a low-rank approximation 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:

        $$
            \begin{equation}
            \psi(\mathbf{w}) = \mathbf{A} \mathbf{B}^\top = \mathbf{W} \in R^{(\sum_{i=1}^k n_i) \times n},
            \end{equation}
        $$
        where $\mathbf{A} \in R^{(\sum_{i=1}^k n_i) \times r}$ and $\mathbf{B} \in R^{n \times r}$ are partitioned and reshaped from the parameter vector $\mathbf{w}$.

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



        Attributes
        ----------
        r : int
            Rank of the low-rank approximation.

        Methods
        -------
        __init__(...)
            Initializes the low-rank parameterized concatenation fusion function.
    """
    def __init__(self, r: int = 2, name: str = 'lowrank_parameterized_concatenation_fusion', *args, **kwargs):
        """
            Initializes the low-rank parameterized concatenation fusion function.

            Parameters
            ----------
            r : int, optional
                Rank of the low-rank approximation. Defaults to 2.
            name : str, optional
                Name of the fusion function. Defaults to "lowrank_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 = lorr_reconciliation(r=self.r)

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

Initializes the low-rank parameterized concatenation fusion function.

Parameters:

Name	Type	Description	Default
r	int	Rank of the low-rank approximation. Defaults to 2.	2
name	str	Name of the fusion function. Defaults to "lowrank_parameterized_concatenation_fusion".	'lowrank_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 = 'lowrank_parameterized_concatenation_fusion', *args, **kwargs):
    """
        Initializes the low-rank parameterized concatenation fusion function.

        Parameters
        ----------
        r : int, optional
            Rank of the low-rank approximation. Defaults to 2.
        name : str, optional
            Name of the fusion function. Defaults to "lowrank_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 = lorr_reconciliation(r=self.r)