  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 not yet been theoretically analyzed.
  This is an example of a promising direction in which theory can inform
machine learning models and machine learning models could inform theory.
