{"id":168057,"date":"2015-03-01T00:00:00","date_gmt":"2015-03-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/multi-k-ic-depth-three-circuit-lower-bound\/"},"modified":"2018-10-16T20:06:18","modified_gmt":"2018-10-17T03:06:18","slug":"multi-k-ic-depth-three-circuit-lower-bound","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/multi-k-ic-depth-three-circuit-lower-bound\/","title":{"rendered":"Multi-k-ic depth three circuit lower bound"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In a multi-<em>k<\/em> -ic depth three circuit every variable appears in at most <em>k<\/em> of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-<em>k<\/em>-ic depth three circuit can be <em>kn<\/em> where <em>n<\/em> is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta, Kamath, Kayal and Saptharishi [GKKS13a] on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-<em>k<\/em> -ic depth three circuits for any arbitrary constant <em>k<\/em> .<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In a multi-k -ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree [&hellip;]<\/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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168057","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"LIPICS","msr_edition":"Symposium on Theoretical Aspects of 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