{"id":168053,"date":"2014-06-01T00:00:00","date_gmt":"2014-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/super-polynomial-lower-bounds-for-depth-four-homogeneous-arithmetic-formulas\/"},"modified":"2018-10-16T20:06:04","modified_gmt":"2018-10-17T03:06:04","slug":"super-polynomial-lower-bounds-for-depth-four-homogeneous-arithmetic-formulas","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/super-polynomial-lower-bounds-for-depth-four-homogeneous-arithmetic-formulas\/","title":{"rendered":"Super-polynomial lower bounds for depth four homogeneous arithmetic formulas"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We show that any depth four homogeneous arithmetic formula computing the Iterated Matrix Multiplication polynomial IMM<sub>n, d<\/sub> \u2014 the (1<em>, <\/em>1)-th entry of the product of <em>d <\/em>generic <em>n-by-<\/em><em>n <\/em>matrices \u2014 has size at least n<sup>log n<\/sup>, if <em>d >= log<\/em><em><sup>2<\/sup><\/em><em> n<\/em> .<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We show that any depth four homogeneous arithmetic formula computing the Iterated Matrix Multiplication polynomial IMMn, d \u2014 the (1, 1)-th entry of the product of d generic n-by-n matrices \u2014 has size at least nlog n, if d >= log2 n .<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"ACM - Association for Computing Machinery","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Symposium on Theory of Computing (STOC)","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"\u00a9 ACM. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. 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