Paper Info
Title
Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
Abstract
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, over fields of zero or large characteristic. 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: - depth-3 (Ω Π Ω) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and - 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 related 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.
Year
DOI
Venue
2011
10.1145/2213977.2214033
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Keywords
DocType
Volume
polynomial time hitting-set generator,depth-3 transcendence degree-k circuit,depth-3 circuit,polynomial time blackbox polynomial,multilinear restriction,common algebraic-geometry theme,bounded top fanin,lower bound,bounded top fanin restriction,identity testing,depth-d occur-k formula,constant-depth constant-read multilinear formula,algebraic geometry,computational complexity,polynomial time,blackbox,depth,jacobian,vandermonde,circuits,model of computation
Journal
abs/1111.0582
Issue
Citations 
PageRank 
4
21
0.67
References 
Authors
45
4
Name
Order
Citations
PageRank
Manindra Agrawal158145.56
Chandan Saha222716.91
Ramprasad Saptharishi318413.72
Nitin Saxena428026.72