Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: (1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (2) constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation

Learning Logistic Circuits

This paper proposes a new classification model called logistic circuits. On MNIST and Fashion datasets, our learning algorithm outperforms neural networks that have an order of magnitude more parameters. Yet, logistic circuits have a distinct origin in symbolic AI, forming a discriminative counterpart to probabilistic-logical circuits such as ACs, SPNs, and PSDDs. We show that parameter learning for logistic circuits is convex optimization, and that a simple local search algorithm can induce strong model structures from data.Comment: Published in the Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI19

Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.Comment: 22 page