{"http:\/\/dx.doi.org\/10.14288\/1.0387498":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Non UBC","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Josh Alman","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2020-01-06T09:39:23Z","type":"literal","lang":"en"},{"value":"2019-07-09T13:32","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"If H is a matrix over a field F, then the rank-r rigidity of H, denoted R_{H}(r), is the minimum Hamming distance from H to a matrix of rank at most r over F. Giving explicit constructions of rigid matrices for a variety of parameter regimes is a central open challenge in complexity theory. In this work, building on Williams' seminal connection between circuit-analysis algorithms and lower bounds [Williams, J. ACM 2014], we give a construction of rigid matrices in P^NP. Letting q = p^r be a prime power, we show:\n\n- There is an absolute constant delta>0 such that, for all constants eps>0, there is a P^NP machine M such that, for infinitely many N's, M(1^N) outputs a matrix H_N in {0,1}^{N times N} with rigidity R_{H_N}(2^{(log N)^{1\/4 - eps}}) >= delta N^2 over F_q.\n\nUsing known connections between matrix rigidity and a number of different areas of complexity theory, we derive several consequences of our constructions, including:\n\n- There is a function f in TIME[2^{(log n)^{omega(1)}}]^NP such that f notin PH^cc. Previously, it was even open whether E^NP subset PH^cc.\n\n- For all eps>0, there is a P^NP machine M such that, for infinitely many N's, M(1^N) outputs an N times N matrix H_N in {0,1}^{N times N} whose linear transformation requires depth-2 F_q-linear circuits of size Omega(N 2^{(log N)^{1\/4 - eps}}). The previous best lower bound for an explicit family of N \\times N matrices was only Omega(N log^2 N \/ log log N), for super-concentrator graphs.\n\nJoint work with Lijie Chen to appear in FOCS 2019.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/73137?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"51.0 minutes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"video\/mp4","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"","type":"literal","lang":"en"},{"value":"Author affiliation: MIT","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/spatial":[{"value":"Banff (Alta.)","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0387498","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus":[{"value":"Unreviewed","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"Banff International Research Station for Mathematical Innovation and Discovery","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/isPartOf":[{"value":"BIRS Workshop Lecture Videos (Banff, Alta)","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/subject":[{"value":"Mathematics","type":"literal","lang":"en"},{"value":"Computer Science, Theoretical Computer Science","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Efficient Construction of Rigid Matrices Using an NP Oracle","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Moving Image","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/73137","type":"literal","lang":"en"}]}}