Learning equivariant tensor functions with applications to sparse vector recovery
Keywords: equivariant machine learning, tensors, sparse vector recovery
TL;DR: We provide a characterization for equivariant tensor polynomials and use that for a machine learning approach to solving the sparse vector recovery problem in new settings.
Abstract: This work characterizes equivariant polynomial functions from tuples of tensor inputs to tensor outputs. Loosely motivated by physics, we focus on equivariant functions with respect to the diagonal action of the orthogonal group on tensors. We show how to extend this characterization to other linear algebraic groups, including the Lorentz and symplectic groups.
Our goal behind these characterizations is to define equivariant machine learning models. In particular, we focus on the sparse vector estimation problem. This problem has been broadly studied in the theoretical computer science literature, and explicit spectral methods, derived by techniques from sum-of-squares, can be shown to recover sparse vectors under certain assumptions. Our numerical results show that the proposed equivariant machine learning models can learn spectral methods that outperform the best theoretically known spectral methods in some regimes. The experiments also suggest that learned spectral methods can solve the problem in settings that have yet to be theoretically analyzed.
This is an example of a promising direction in which theory can inform machine learning models and machine learning models can inform theory.
Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
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Submission Number: 12216
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