Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs)

Algebraic Methods in Computational Complexity

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. In some of the most exciting recent progress in Computational Complexity the algebraic theme still plays a central role. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Also the areas of derandomization and coding theory have experimented important advances. The seminar aimed to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and the goal of the seminar was to play an important role in educating a diverse community about the latest new techniques

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support “cryptography” secure against algebraic circuits. Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem, that is, to the existence of a hitting set for the class of poly(N)-degree poly(N)-size circuits which consists of coefficient vectors of polynomials of polylog(N) degree with polylog(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni (2001), except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits. A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017)

On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for tensors of degree 2 (matrices), and obtain quasi-polynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank recovery asks (say, over the reals) to recover a matrix M from few measurements, under the promise that M is rank <=r. We also give a formal connection between low-rank recovery and the task of sparse (vector) recovery: any sparse-recovery algorithm that exactly recovers vectors of length n and sparsity 2r, using m non-adaptive measurements, yields a low-rank recovery scheme for exactly recovering nxn matrices of rank <=r, making 2nm non-adaptive measurements. Furthermore, if the sparse-recovery algorithm runs in time \tau, then the low-rank recovery algorithm runs in time O(rn^2+n\tau). We obtain this reduction using linear-algebraic techniques, and not using convex optimization, which is more commonly seen in compressed sensing algorithms. By using a dual Reed-Solomon code, we are able to (deterministically) construct low-rank recovery schemes taking 4nr measurements over the reals, such that the measurements can be all rank-1 matrices, or all sparse matrices.Comment: 55 page