The limits of depth reduction for arithmetic formulas: It's all about the top fan-in

Mrinal Kumar, Shubhangi Saraf

Research output: Chapter in Book/Report/Conference proceedingConference contribution

24 Scopus citations

Abstract

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal and Vinay[1], Koiran [11] and Tavenas [16], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [9] and Gupta et al [5]. These results inspired a urry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [10] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial lower bounds for general (homogeneous) arithmetic formulas, and if so this would indeed be a breakthrough in arithmetic circuit complexity. In this paper we study the power and limitations of depth reduction and shifted partial derivatives for arithmetic formulas. We do it via studying the class of depth 4 homogeneous arithmetic circuits. We show: (1) the first superpolynomial lower bounds for the class of homogeneous depth 4 circuits with top fan-in o(log n). The core of our result is to show improved depth reduction for these circuits. This class of circuits has received much attention for the problem of polynomial identity testing. We give the first nontrivial lower bounds for these circuits for any top fan-in ≥ 2. (2) We show that improved depth reduction is not possible when the top fan-in isω(log n). In particular this shows that the depth reduction procedure of Koiran and Tavenas [11, 16] cannot be improved even for homogeneous formulas, thus strengthening the results of Fournier et al [3] who showed that depth reduction is tight for circuits, and answering some of the main open questions of [10, 3]. Our results in particular suggest that the method of improved depth reduction and shifted partial derivatives may not be powerful enough to prove superpolynomial lower bounds for (even homogeneous) arithmetic formulas.

Original languageEnglish (US)
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages136-145
Number of pages10
ISBN (Print)9781450327107
DOIs
StatePublished - 2014
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: May 31 2014Jun 3 2014

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other4th Annual ACM Symposium on Theory of Computing, STOC 2014
Country/TerritoryUnited States
CityNew York, NY
Period5/31/146/3/14

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Arithmetic formula
  • Depth reduction
  • Lower bounds
  • Shifted partial derivatives

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