License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.78
URN: urn:nbn:de:0030-drops-106548
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10654/
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Kumar, Mrinal ; Oliveira, Rafael ; Saptharishi, Ramprasad

Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

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Abstract

We show that any n-variate polynomial computable by a syntactically multilinear circuit of size poly(n) can be computed by a depth-4 syntactically multilinear (Sigma Pi Sigma Pi) circuit of size at most exp ({O (sqrt{n log n})}). For degree d = omega(n/log n), this improves upon the upper bound of exp ({O(sqrt{d}log n)}) obtained by Tavenas [Sébastien Tavenas, 2015] for general circuits, and is known to be asymptotically optimal in the exponent when d < n^{epsilon} for a small enough constant epsilon. Our upper bound matches the lower bound of exp ({Omega (sqrt{n log n})}) proved by Raz and Yehudayoff [Ran Raz and Amir Yehudayoff, 2009], and thus cannot be improved further in the exponent. Our results hold over all fields and also generalize to circuits of small individual degree.
More generally, we show that an n-variate polynomial computable by a syntactically multilinear circuit of size poly(n) can be computed by a syntactically multilinear circuit of product-depth Delta of size at most exp inparen{O inparen{Delta * (n/log n)^{1/Delta} * log n}}. It follows from the lower bounds of Raz and Yehudayoff [Ran Raz and Amir Yehudayoff, 2009] that in general, for constant Delta, the exponent in this upper bound is tight and cannot be improved to o inparen{inparen{n/log n}^{1/Delta}* log n}.

BibTeX - Entry

@InProceedings{kumar_et_al:LIPIcs:2019:10654,
  author =	{Mrinal Kumar and Rafael Oliveira and Ramprasad Saptharishi},
  title =	{{Towards Optimal Depth Reductions for Syntactically Multilinear Circuits}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{78:1--78:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10654},
  URN =		{urn:nbn:de:0030-drops-106548},
  doi =		{10.4230/LIPIcs.ICALP.2019.78},
  annote =	{Keywords: arithmetic circuits, multilinear circuits, depth reduction, lower bounds}
}

Keywords: arithmetic circuits, multilinear circuits, depth reduction, lower bounds
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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