{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T03:36:33Z","timestamp":1789443393089,"version":"build-2803163510"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth-4 arithmetic formula computing the product of $d$ generic matrices of size $n \\times n$, $\\mathrm{IMM}_{n,d}$, has size $n^{\\Omega(\\sqrt{d})}$ as long as $d = n^{O(1)}$. This improves the result of Nisan and Wigderson [Comput. Complexity, 6 (1997), pp. 217--234] for depth-4 set-multilinear formulas. We also study $\\Sigma\\Pi^{[O(d\/t)]}\\Sigma\\Pi^{[t]}$ formulas, which are depth-4 formulas with the stated bounds on the fan-ins of the $\\Pi$ gates. A recent depth reduction result of Tavenas [Lecture Notes in Comput. Sci. 8087, 2013, pp. 813--824] shows that any $n$-variate degree $d = n^{O(1)}$ polynomial computable by a circuit of size $\\mathop{\\mathrm{poly}}(n)$ can also be computed by a depth-4 $\\Sigma\\Pi^{[O(d\/t)]}\\Sigma\\Pi^{[t]}$ formula of top fan-in $n^{O(d\/t)}$. We show that any such formula computing $\\mathrm{IMM}_{n,d}$ has top fan-in $n^{\\Omega({d\/t})}$, proving the optimality of Tavenas' result. This also strengthens a result of Kayal, Saha, and Saptharishi [Proceedings of STOC, 2014, pp. 146--153], which gives a similar lower bound for an explicit polynomial in VNP.<\/jats:p>","DOI":"10.1137\/140990280","type":"journal-article","created":{"date-parts":[[2015,9,8]],"date-time":"2015-09-08T11:42:01Z","timestamp":1441712521000},"page":"1173-1201","source":"Crossref","is-referenced-by-count":11,"title":["Lower Bounds for Depth-4 Formulas Computing Iterated Matrix Multiplication"],"prefix":"10.1137","volume":"44","author":[{"given":"Herv\u00e9","family":"Fournier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nutan","family":"Limaye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guillaume","family":"Malod","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Srikanth","family":"Srinivasan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,9,8]]},"reference":[{"key":"atypb1","first-page":"67","author":"Agrawal M.","year":"2008","journal-title":"Proceedings of FOCS"},{"key":"atypb2","unstructured":"D. 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