Hitting sets for multilinear read-once algebraic branching programs, in any order

Abstract

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n O(lg 2 n) time.Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order.We obtain our results by recasting, and improving upon, the ideas of Agrawal, Saha and Saxena [ASS13]. We phrase the ideas in terms of rank condensers and Wronskians, and show that our results improve upon the classical multivariate Wronskian, which may be of independent interest.In addition, we give the first n O(lg lg n) black-box polynomial identity testing algorithm for the so called model of diagonal circuits. This result improves upon the n Θ(lg n) -time algorithms given by Agrawal, Saha and Saxena [ASS13], and Forbes and Shpilka [FS13b] for this class. More generally, our result holds for any model computing polynomials whose partial derivatives (of all orders) span a low dimensional linear space.

Geometric complexity theory V: Efficient algorithms for Noether normalization

“…This was the result that was stated in the preliminary version [69] of this article. The stronger O(n O(log log n) )-time bound stated in Theorem 1.6 follows in view of the recent result in Forbes, Saptharishi, and Shpilka [29], which gives an O(s O(log log s) )-time-computable black-box derandomization of polynomial identity testing for diagonal depth three circuits of size ≤ s.…”

“…The formal notion of explicitness introduced in this article (Definition 5.1) lies at the heart of the proofs, along with the fundamental work [43,75,78,16,80,82,32,57] in algebraic geometry and geometric invariant theory, the fundamental work [91,93,88,50,51,52,60] in algebraic complexity theory, and the fundamental work [41,76,47,49,86,2,30,29] on a derandomization problem in complexity theory, called black-box polynomial identity testing. Derandomization means converting a randomized efficient algorithm into a deterministic efficient algorithm by removing random bits.…”

Testing Equivalence of Polynomials under Shifts

“…Their result, together with our algorithm, imply that we can find out whether two ROABPs are shift-equivalent in deterministic quasi-polynomial time. Since this class also captures tensors, 1 an application of our result is that we can find out whether two tensors are shift-equivalent in quasi-polynomial time (we refer the reader to [FS13,FSS13] for definitions of ROABPs and tensors). Corollary 1.8.…”

“…It is clear that solving PIT in the black-box model is at least as hard as solving it in the white-box model and indeed, in some cases we have better algorithms in the white-box model than in the black-box model (compare e.g. [RS05] to [FS13] and [FSS13]). …”

“…In the past decade a series of algorithms [DS06, KS07, KS11, SS11, KS09b, SS10, ASSS12] were devised to solve PIT for circuits of depth 3 with bounded fan-in, which are denoted as ΣΠΣ(k) circuits. A more recent line of work, to which we will go back later in the paper, deals with read once branching programs and low rank tensors [FS12,FS13,FSS13].…”