Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in

Abstract

In this paper we give reconstruction algorithms for depth-3 arithmetic circuits with k multiplication gates (also known as ΣΠΣ(k) circuits), where k = O(1). Namely, we give an algorithm that when given a black box holding a ΣΠΣ(k) circuit C over a field F as input, makes queries to the black box (possibly over a polynomial sized extension field of F) and outputs a circuit C computing the same polynomial as C. In particular we obtain the following results.1. When C is a multilinear ΣΠΣ(k) circuit (i.e. each of its multiplication gates computes a multilinear polynomial) then our algorithm runs in polynomial time (when k is a constant) and outputs a multilinear ΣΠΣ(k) circuits computing the same polynomial.2. In the general case, our algorithm runs in quasi polynomial time and outputs a generalized depth-3 circuit (a notion that is defined in the paper) with k multiplication gates. In fact, this algorithm works in the slightly more general case where the black box holds a generalized depth-3 circuits.Prior to this work there were reconstruction algorithms for several different models of bounded depth circuits: the well studied class of depth-2 arithmetic circuits (that compute sparse polynomials) and its close by model of depth-3 set-multilinear circuits. For the class of depth-3 circuits only the case of k = 2 (i.e. ΣΠΣ(2) circuits) was known.Our proof technique combines ideas from [Shp07] and [KS08] with some new ideas. Our most notable new ideas are: We prove the existence of a unique canonical representation of depth-3 circuits. This enables us to work with a specific representation in mind. Another technical contribution is an isolation lemma for depth-3 circuits that enables us to reconstruct a single multiplication gate of the circuit.

Arithmetic Circuits: A survey of recent results and open questions

“…Currently, reconstruction algorithms are known for depth-2 circuits, see Refs. [GK87, BT88, GKS90, KS01] and references within, for depth-3 circuits with bounded top fan-in [Shp09,KS09a] and for set-multilinear noncommutative formulas [BBB + 00, KS06]. It is an interesting question whether there is a generic way of transforming a black-box PIT algorithm to a reconstruction algorithm.…”

“…First we will explain the idea behind the algorithm of Shpilka [Shp09] that works for circuits with two multiplication gates (k = 2) and then the extension of Ref. [KS09a] that works for the case of k = O(1) multiplication gates.…”

“…We now explain the idea behind the algorithm of [KS09a] for reconstructing ΣΠΣ(k) circuits. Before we describe their algorithm let us explain the difficulties in extending the algorithm above for the case k > 2.…”

“…Basically, this technique can be thought of as learning via partial derivatives. At the end, we move to depth-3 circuits with a bounded top fan-in and sketch the algorithms of [Shp09,KS09a] that are based on ideas from the identity testing algorithm of [DS06,KS09a].…”

Testing Equivalence of Polynomials under Shifts

“…We also note that we can make their algorithm work in the black-box case as well. Using the reconstruction algorithms of [Shp09,KS09a] we can first reconstruct the ΣΠΣ(k) circuit in quasi-polynomial time. We can also interpolate the sparse polynomial in polynomial time (for interpolation of sparse polynomials see e.g.…”