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Towards Optimal Depth Reductions for Syntactically Multilinear Circuits
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Colloquium on Automata, Languages, and Programming (ICALP) - License:
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- Publication Date: 2019-07-04
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Acknowledgements
We are extremely thankful to Ben Rossman, who pointed us towards this question, and for many stimulating discussions at various stages of this work. We also thank Shubhangi Saraf, Amir Shpilka and Ben Lee Volk for many helpful conversations. Mrinal also thanks Prahladh Harsha for accommodating him in his apartment for a part of the visit to TIFR, where a part of this paper was written.
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Mrinal Kumar, Rafael Oliveira, and Ramprasad Saptharishi. Towards Optimal Depth Reductions for Syntactically Multilinear Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.78
https://doi.org/10.4230/LIPIcs.ICALP.2019.78
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}.
Subject Classification
ACM Subject Classification
- Theory of computation → Circuit complexity
Keywords
- arithmetic circuits
- multilinear circuits
- depth reduction
- lower bounds
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References
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