Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

Abstract

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity n O(log n) . In both the cases, our time complexity is double exponential in the number of ROABPs.ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time.Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

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“…Our construction of s(t 1 ,t 2 ) comes from the basis isolating weight assignment for ROABPs from Agrawal et al [2]. We use the fact that for any polynomial over a k-dimensional algebra, shift by a basis isolating map achieves log(k + 1)-concentration [17].…”

“…Gurjar et al [17,Lemma 5.2] have shown that shifting by a basis isolating weight assignment achieves concentration. We write their lemma here without a proof.…”

“…However, we want a single tuple s which works for every D(x). To get a single tuple, we combine the tuples in {t w } w∈W using the standard technique of Lagrange interpolation (also used in [14,17]). Let {α w } w∈W be distinct constants.…”

Identity Testing and Lower Bounds for Read- k Oblivious Algebraic Branching Programs

“…Indeed, in the boolean domain, pseudorandom generators for read-once branching programs in unknown order are much weaker, in terms of the seed length, than Nisan's generator [Nis92] which works only if the order is known. Recently, Gurjar et al obtained PIT algorithms for sum of ROABPs [GKST15].…”

“…Some results were known for the more restricted model of a sum of k ROABPs (e.g. [GKST15]), and we give more details on those in Section 1.3.…”