funfact 0.6

Functional factorization for matrices and tensors

FunFact

FunFact is a library for computing the functional factorization of algebraic tensors, a.k.a. multidimensional arrays. A functional factorization, in our context, is a generalization of the (linear) factorization of tensors. By generalization, we meant to replace the standard inner/outer product between the factor tensors with nonlinear operations.

For example, a rank-1 matrix can be factored into the outer product between a column vector and a row vector:

.. math::

M \approx \mathbf{u} \mathbf{v}^\mathsf{T},

where :math:M is an :math:n \times m matrix, :math:\mathbf{u} is a :math:n-dimensional column vector, and :math:\mathbf{v} is a :math:m-dimensional row vector. This can be equivalently represented in indexed notation as

.. math::

M_{ij} \approx \mathbf{u}_i \mathbf{v}_j.

Moreover, if we relace the standard multiplication operation between :math:\mathbf{u}_i and :math:\mathbf{v}_j by an RBF function :math:\kappa(x, y) = \exp\left[-(x - y)^2\right], we then obtain an RBF approximation <https://arxiv.org/abs/2106.02018>__ of :math:M such that:

.. math::

M_{ij} \approx \kappa(\mathbf{u}_i, \mathbf{v}_j).

Given the rich expressivity of nonlinear operators and functional forms, we expect that a proper functional factorization of a tensor can yield representations that are more compact than what is possible withtin the existing linear framework. However, there is (obviously) no free lunch. the challenges to obtain the functional factorization of a tensor is two fold and involves - Finding the most appropriate functional form given a specific piece of data, - Finding the component tensors given the functional form for a specific data.

The two points above are exactly what we aim to facilitate using FunFact.

Copyright

FunFact Copyright (c) 2021, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from the U.S. Dept. of Energy). All rights reserved.

If you have questions about your rights to use or distribute this software, please contact Berkeley Lab’s Intellectual Property Office at [e-mail].

NOTICE. This Software was developed under funding from the U.S. Department of Energy and the U.S. Government consequently retains certain rights. As such, the U.S. Government has been granted for itself and others acting on its behalf a paid-up, nonexclusive, irrevocable, worldwide license in the Software to reproduce, distribute copies to the public, prepare derivative works, and perform publicly and display publicly, and to permit others to do so.