Learning Polynomial Problems with $SL(2,\mathbb{R})$ Equivariance
Hannah Lawrence Mitchell Tong Harris
Optimizing and certifying the positivity of polynomials are fundamental
primitives across mathematics and engineering applications, from dynamical
systems to operations research. However, solving these problems in practice
requires large semidefinite programs, with poor scaling in dimension and
degree. In this work, we demonstrate for the first time that neural networks
can effectively solve such problems in a data-driven fashion, achieving tenfold
speedups while retaining high accuracy. Moreover, we observe that these
polynomial learning problems are equivariant to the non-compact group
$SL(2,\mathbb{R})$, which consists of area-preserving linear transformations.
We therefore adapt our learning pipelines to accommodate this structure,
including data augmentation, a new $SL(2,\mathbb{R})$-equivariant architecture,
and an architecture equivariant with respect to its maximal compact subgroup,
$SO(2, \mathbb{R})$. Surprisingly, the most successful approaches in practice
do not enforce equivariance to the entire group, which we prove arises from an
unusual lack of architecture universality for $SL(2,\mathbb{R})$ in particular.
A consequence of this result, which is of independent interest, is that there
exists an equivariant function for which there is no sequence of equivariant
polynomials multiplied by arbitrary invariants that approximates the original
function. This is a rare example of a symmetric problem where data augmentation
outperforms a fully equivariant architecture, and provides interesting lessons
in both theory and practice for other problems with non-compact symmetries.