Diagonal Circuit Identity Testing and Lower Bounds

Abstract

In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth-3 circuit C(x ₁,...,x _n ) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results:

• 1 Suppose we are given a depth-4 circuit (over any field $\mathbb{F}$) of the form:

  $$C({x_1},\ldots,{x_n}):=\sum_{i=1}^k L_{i,1}^{e_{i,1}}\cdots L_{i,s}^{e_{i,s}}$$

  where, each L _i,j is a sum of univariate polynomials in $\mathbb{F}[{x_1},\ldots,{x_n}]$. We can test whether C is zero deterministically in poly(size(C), max _i {(1 + e _i,1) ⋯ (1 + e _i,s)}) field operations. In particular, this gives a deterministic polynomial time identity test for general depth-3 circuits C when the d: =degree(C) is logarithmic in the size(C).

• 1 We prove that if the above circuit C(x ₁,...,x _n ) computes the determinant (or permanent) of an m×m formal matrix with a “small” $s=o\left(\frac{m}{\log m}\right)$ then k = 2^Ω(m). Our lower bounds work for all fields $\mathbb{F}$. (Previous exponential lower bounds for depth-3 only work for nonzero characteristic.)

• 1 We also present an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(nklog(d)) field operations over finite fields).