What is the Σ₃²-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2^{α_n n} for some α_n = Ω(1). We show that the limiting constant α := lim sup α_n satisfies 0.847... ≤ α ≤ 0.965... . Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that α ∈ [0.5,1]. To obtain our improved bounds, we analyse a covering LP that captures the Σ₃²-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.
@InProceedings{goos_et_al:LIPIcs.MFCS.2023.51, author = {G\"{o}\"{o}s, Mika and Guan, Ziyi and Mosnoi, Tiberiu}, title = {{Depth-3 Circuits for Inner Product}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.51}, URN = {urn:nbn:de:0030-drops-185856}, doi = {10.4230/LIPIcs.MFCS.2023.51}, annote = {Keywords: Circuit complexity, inner product} }
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