2011/11/02 by Manindra Agrawal, Chandan Saha, Agrawal, Manindra +5
Computer Science · Mathematics · #68W30 #Adversarial Robustness in Machine Learning #Commutative Algebra (math.AC) #Computational Complexity (cs.CC) #F.2.1 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning and Algorithms #VLSI and Analog Circuit Testing #acm:68W30 #cs.CC #math.AC #msc:68W30
paper · pdf · doi:10.48550/arxiv.1111.0582
arxiv created 2011/11/02 · openalex publication_date 2011/11/02 · arxiv updated 2011/11/03 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
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.