Abstract
We introduce the polynomial coefficient matrix and identify the maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results:
—As our first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n × n requires Ω(nd − 1/2d) size. This improves the lower bounds in Nisan and Wigderson [1995] for d = ω(1).
—As our second main result, we show that there is an explicit polynomial on n variables and degree at most n/2 for which any depth-3 circuit of product dimension at most n/10 (dimension of the space of affine forms feeding into each product gate) requires size 2Ω(n). This generalizes the lower bounds against diagonal circuits proved in Saxena [2008]. Diagonal circuits are of product dimension 1.
—We prove a nΩ(log n) lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, this result extends the known super-polynomial lower bounds on the size of multilinear formulas [Raz 2006].
—We prove a 2Ω(n) lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs [Jansen 2008].
- Manindra Agrawal and V. Vinay. 2008. Arithmetic circuits: A chasm at depth four. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 67--75. Google Scholar
Digital Library
- Zeev Dvir, Guillaume Malod, Sylvain Perifel, and Amir Yehudayoff. 2012. Separating multilinear branching programs and formulas. In Proceedings of the Symposium on Theory of Computing (STOC). ACM Special Interest Group on Algorithms and Computation Theory, 615--624. Google Scholar
Digital Library
- Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. 2013. Lower bounds for depth 4 formulas computing iterated matrix multiplication. Electronic Colloquium on Computational Complexity (ECCC) 20 (2013), 100.Google Scholar
- Dima Grigoriev and Marek Karpinski. 1998. An exponential lower bound for depth 3 arithmetic circuits. In Proceedings of the Symposium on Theory of Computing (STOC). ACM Special Interest Group on Algorithms and Computation Theory, 577--582. Google Scholar
Digital Library
- D. Grigoriev and A. Razborov. 1998. Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 269--278. Google Scholar
Digital Library
- A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. 2013a. Approaching the chasm at depth four. In Proceedings of the CCC (2013). IEEE Computer Society.Google Scholar
- Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. 2013b. Arithmetic circuits: A chasm at depth three. In Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS 2015). IEEE Computer Society, 578--587. Google Scholar
Digital Library
- Maurice J. Jansen. 2008. Lower bounds for syntactically multilinear algebraic branching programs. In Proceedings of International Symposium on Mathematical Foundations of Computer Science (MFCS). Springer, 407--418. Google Scholar
Digital Library
- Neeraj Kayal. 2012. An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC) 19 (2012), 81.Google Scholar
- Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. 2014. An exponential lower bound for homogeneous depth four arithmetic formulas. Electronic Colloquium on Computational Complexity (ECCC) (2014).Google Scholar
- Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. 2013. A super-polynomial lower bound for regular arithmetic formulas. Electronic Colloquium on Computational Complexity (ECCC) 20 (2013), 91.Google Scholar
- Pascal Koiran. 2012. Arithmetic circuits: The chasm at depth four gets wider. Theoretical Computer Science 448 (2012), 56--65. Google Scholar
Digital Library
- Mrinal Kumar and Shubhangi Saraf. 2013a. Lower bounds for depth 4 homogenous circuits with bounded top fanin. Electronic Colloquium on Computational Complexity (ECCC) 20 (2013), 68.Google Scholar
- Mrinal Kumar and Shubhangi Saraf. 2013b. Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits. CoRR abs/1312.5978 (2013).Google Scholar
- N. Nisan and A. Wigderson. 1995. Lower bounds on arithmetic circuits via partial derivatives. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 16--25. Google Scholar
Digital Library
- Ran Raz. 2006. Separation of multilinear circuit and formula size. Theory of Computing 2, 1 (2006), 121--135.Google Scholar
Cross Ref
- Ran Raz. 2009. Multi-linear formulas for permanent and determinant are of super-polynomial size. Journal of the ACM 56, Article 8 (April 2009), 17 pages. Issue 2. Google Scholar
Digital Library
- Ran Raz, Amir Shpilka, and Amir Yehudayoff. 2008. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM Journal of Computing 38, 4 (2008), 1624--1647. Google Scholar
Digital Library
- R. Raz and A. Yehudayoff. 2008. Lower bounds and separations for constant depth multilinear circuits. In Proceedings of the Conference on Computational Complexity. 128--139. Google Scholar
Digital Library
- Nitin Saxena. 2008. Diagonal circuit identity testing and lower bounds. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP). Springer, 60--71. Google Scholar
Digital Library
- Amir Shpilka. 2001. Affine projections of symmetric polynomials. In Proceedings of the Conference on Computational Complexity. IEEE Computer Society, 160--171. Google Scholar
Digital Library
- Amir Shpilka and Avi Wigderson. 2001. Depth-3 arithmetic circuits over fields of characteristic zero. Computational Complexity 10, 1 (2001), 1--27. Google Scholar
Digital Library
- Amir Shpilka and Amir Yehudayoff. 2010. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5, 3--4 (March 2010), 207--388. DOI:http://dx.doi.org/10.1561/0400000039 Google Scholar
Digital Library
- Sébastien Tavenas. 2013. Improved bounds for reduction to depth 4 and depth 3. In Proceedings of International Symposium on Mathematical Foundations of Computer Science (MFCS). Springer, 813--824.Google Scholar
Cross Ref
Index Terms
Arithmetic Circuit Lower Bounds via Maximum-Rank of Partial Derivative Matrices
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