Rafael Mendes de Oliveira

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(nδ), we give a hitting set of size exp O (n2/3+2δ/3)). This implies a lower bound of exp [EQUATION] for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(nδ), we give a hitting set of size exp [EQUATION]. This implies a lower bound of exp [EQUATION] for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +, x gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp (n1-δ), for regular depth-d multilinear formulas of size exp(nδ), where [EQUATION]. This result implies a lower bound of roughly exp [EQUATION] 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).

Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via operator scaling

The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous in- equalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is known about computing their main parameters below (which we later define). Prior to this work, the best known algorithms for any of these optimization tasks required at least exponential time. In this work, we give polynomial time algorithms to compute: (1) Feasibility of BL-datum, (2) Optimal BL- constant, (3) Weak separation oracle for BL-polytopes. What is particularly exciting about this progress, beyond the better understanding of BL- inequalities, is that the objects above naturally encode rich families of optimization problems which had no prior efficient algorithms. In particular, the BL-constants (which we efficiently compute) are solutions to non-convex optimization problems, and the BL-polytopes (for which we provide efficient membership and separation oracles) are linear programs with exponentially many facets. Thus we hope that new combinatorial optimization problems can be solved via reductions to the ones above, and make modest initial steps in exploring this possibility. Our algorithms are obtained by a simple efficient reduction of a given BL-datum to an instance of the Operator Scaling problem defined by [Gur04]. To obtain the results above, we utilize the two (very recent and different) algorithms for the operator scaling problem [GGOW16, IQS15a]. Our reduction implies algorithmic versions of many of the known structural results on BL-inequalities, and in some cases provide proofs that are different or simpler than existing ones. Further, the analytic properties of the [GGOW16] algorithm provide new, effective bounds on the magnitude and continuity of BL-constants, with applications to non-linear versions of BL-inequalities; prior work relied on compactness, and thus provided no bounds. On a higher level, our application of operator scaling algorithm to BL-inequalities further connects analysis and optimization with the diverse mathematical areas used so far to mo- tivate and solve the operator scaling problem, which include commutative invariant theory, non-commutative algebra, computational complexity and quantum information theory.

Testing Equivalence of Polynomials under Shifts

Two polynomials f;g2 F[x1;:::;xn] are called shift-equivalent if there exists a vector (a1;:::;an)2 F n such that the polynomial identity f(x1 +a1;:::;xn +an) g(x1;:::;xn) holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev [Gri97] who gave a deterministic algorithm running in time n O(d) for degree d polynomials. Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

Much Faster Algorithms for Matrix Scaling

We develop several efficient algorithms for the classical Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input n× n matrix A, this problem asks to find diagonal (scaling) matrices X and Y (if they exist), so that X A Y ε-approximates a doubly stochastic matrix, or more generally a matrix with prescribed row and column sums.We address the general scaling problem as well as some important special cases. In particular, if A has m nonzero entries, and if there exist X and Y with polynomially large entries such that X A Y is doubly stochastic, then we can solve the problem in total complexity \tilde{O}(m + n^{4/3}). This greatly improves on the best known previous results, which were either \tilde{O}(n^4) or O(m n^{1/2}/ε).Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems.

Factors of low individual degree polynomials

In [8], Kaltofen proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofens work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in [10], and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if P(x1,..., xn) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor f(x1,..., xn) of P (x1,..., xn) can be computed by a formula of size poly((rn)r, s) and depth d+5. This partially answers the question above posed in [10], that asked if this result holds without the exponential dependence on r. Our work generalizes the main factorization theorem from Dvir et al. [2], who proved it for the special case when the factors are of the form f(x1,..., xn) ≡ xn ---g(x1,..., xn−1). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).

Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory

Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently. In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem. This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling. Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems. Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT).

Locally testable and locally correctable codes approaching the gilbert-varshamov bound

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance Δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every Δ ∈ (0, 1/2), there are codes with minimum distance Δ and rate R = RGV (Δ) > 0 (for a certain simple function RGV(·)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1. Local Testing: For all Δ ∈ (0, 1/2) and all R 2. Local Correction: For all ϵ > 0, for all Δ > 1/2 sufficiently large, and all R Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesens random concatenation technique and the best known locally testable codes from [KMRS16]. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, Alon-Edmonds-Luby distance amplification, and the local list-decodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.

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

Rank bounds for design matrices with block entries and geometric applications

{(n^\delta)}