Lower bounds for matrix product, in bounded depth circuits with arbitrary gates

Abstract

We prove super-linear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bounds apply for several explicit functions, and, most importantly, for matrix product. In particular, we obtain the following results:1. We show that the number of edges in any constant depth arithmetic circuit for matrix product (over any field) is super-linear in m 2 (where m × m is the size of each matrix). That is, the lower bound is superlinear in the number of input variables. Moreover, if the circuit is bilinear the result applies also for the case where the circuit gets for free any product of two linear functions.2. We show that the number of edges in any constant depth arithmetic circuit for the trace of the product of 3 matrices (over fields with characteristic 0) is superlinear in m 2 . (Note that the trace is a single-output function).3. We give explicit examples for n Boolean functions f1, . . . , fn, such that any constant depth Boolean circuit with arbitrary gates for f1, ..., fn has a superlinear number of edges. The lower bound is proved also for circuits with arbitrary gates over any finite field. The bound applies for matrix product over finite fields as well as for several other explicit functions.

A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

“…We consider bounded depth bilinear arithmetical circuits with arbitrary fan-in and fan-out as in [RS03]. Constants on the wires are assumed to be from the complex field C, and our circuits are assumed to be layered.…”

“…In this paper, building on the work of [RS03], we prove a size-depth tradeoff for the circular convolution mapping that was considered in [BL02]. We do this utilizing a discrete variant of the Heisenberg uncertainty principle.…”

“…Although we do not know of any non-linear lower bounds for unrestricted linear and bilinear circuits, what is known are size-depth tradeoffs. Namely, for linear circuits there are the results by Pudlak [Pud94], and for bilinear circuits of constant depth Raz and Shpilka prove a non-linear lower bound for the matrix multiplication mapping [RS03].…”

Circuits with Arbitrary Gates

“…Super-linear lower bounds up to s 2 (f ) = Ω(n log 2 n) where proved using graph-theoretic arguments by analyzing some super-concentration properties of the circuit as a graph [5,9,10,12,11,1,13,14,15]. Higher lower bounds of the form s 2 (f ) = Ω(n 3/2 ) were recently proved using information theoretical arguments [4,6].…”

Computing Elementary Symmetric Polynomials with a Subpolynomial Numberof Multiplications

“…Most recently, Raz and Shpilka got nice lower bound results for constant-depth arithmetic circuits [RS01], and Raz [Raz02] proved a Ω(n 2 log n) lower bound for matrix-multiplication in this model, solving a long-standing open problem.…”