Abstract
Koiran [7] showed that if an n-variate polynomial of degree d (with d = n O(1)) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size \(2^{O(\sqrt{d}\log(d)\log(s))}\). Using this result, Gupta, Kamath, Kayal and Saptharishi [6] gave an exp \(\left(O\left(\sqrt{d\log(d)\log(n)\log(s)}\right)\right)\) upper bound for the size of the smallest depth three circuit computing an n-variate polynomial of degree d = n O(1) given by a circuit of size s.
We improve here Koiran’s bound. Indeed, we show that if we reduce an arithmetic circuit to depth four, then the size becomes exp\(\left(O\left(\sqrt{d\log(ds)\log(n)}\right)\right)\). Mimicking the proof in [6], it also implies the same upper bound for depth three circuits.
This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal and Saptharishi [5] also showed a \(2^{\Omega(\sqrt{d})}\) lower bound for the size of homogeneous depth four circuits such that gates at the bottom have fan-in at most \(\sqrt{d}\). Finally, we show that this last lower bound also holds if the fan-in is at least \(\sqrt{d}\).
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References
Agrawal, M., Vinay, V.: Arithmetic circuits: A chasm at depth four. In: Proceedings-Annual Symposium on Foundations of Computer Science, pp. 67–75 (2008)
Allender, E., Jiao, J., Mahajan, M., Vinay, V.: Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science 209(1-2), 47–86 (1998)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer (2000)
Chen, X., Kayal, N., Wigderson, A.: Partial Derivatives in Arithmetic Complexity and Beyond. Foundations and Trends in Theoretical Computer Science (2011)
Gupta, A., Kamath, P., Kayal, N., Saptharishi, R.: Approaching the chasm at depth four. In: Proceedings of the Conference on Computational Complexity, CCC (2013)
Gupta, A., Kamath, P., Kayal, N., Saptharishi, R.: Arithmetic circuits: A chasm at depth three. Electronic Colloquium on Computational Complexity (2013)
Koiran, P.: Arithmetic circuits: The chasm at depth four gets wider. Theoretical Computer Science 448, 56–65 (2012)
Shpilka, A., Yehudayoff, A.: Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, vol. 5 (2010)
Valiant, L., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM Journal on Computing 12(4), 641–644 (1983)
von zur Gathen, J.: Feasible arithmetic computations: Valiant’s hypothesis. Journal of Symbolic Computation 4(2), 137–172 (1987)
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Tavenas, S. (2013). Improved Bounds for Reduction to Depth 4 and Depth 3. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_71
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DOI: https://doi.org/10.1007/978-3-642-40313-2_71
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