The Power of Depth 2 Circuits over Algebras

TL;DR

This paper explores polynomial identity testing for depth 2 circuits over matrix algebras, establishing equivalences with higher-depth circuits and providing efficient algorithms for certain algebraic structures.

Contribution

It demonstrates the polynomial-time equivalence between depth 3 circuits over a field and depth 2 circuits over upper-triangular matrices, and offers a deterministic algorithm for constant-dimensional commutative algebras.

Findings

PIT of depth 3 circuits reduces to PIT of width-2 planar ABPs.

Deterministic polynomial-time PIT for Pi-Sigma circuits over constant-dimensional commutative algebras.

Pi-Sigma circuits over polynomial-dimensional commutative algebras are as hard as Sigma-Pi-Sigma circuits.

Abstract

We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (Pi-Sigma) arithmetic circuits over U_2(F), the algebra of upper-triangular 2 x 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, Pi-Sigma circuits over U_2(F) are strictly `weaker' than Sigma-Pi-Sigma circuits over F. The equivalence further shows that PIT of depth 3 arithmetic circuits reduces to PIT of width-2 planar commutative Algebraic Branching Programs (ABP). Thus, identity testing for commutative ABPs is interesting even in the case of width-2. Further, we give a deterministic polynomial time identity testing algorithm for a Pi-Sigma circuit over any constant dimensional commutative algebra over F. While over commutative algebras of polynomial dimension, identity testing is at least as hard as that of Sigma-Pi-Sigma circuits over F.