{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","Extent":"http:\/\/purl.org\/dc\/terms\/extent","FileFormat":"http:\/\/purl.org\/dc\/elements\/1.1\/format","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","GeographicLocation":"http:\/\/purl.org\/dc\/terms\/spatial","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Notes":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","PeerReviewStatus":"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","RightsURI":"https:\/\/open.library.ubc.ca\/terms#rightsURI","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Series":"http:\/\/purl.org\/dc\/terms\/isPartOf","Subject":"http:\/\/purl.org\/dc\/terms\/subject","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Non UBC","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Creator":[{"@value":"Josh Alman","@language":"en"}],"DateAvailable":[{"@value":"2020-01-06T09:39:23Z","@language":"en"}],"DateIssued":[{"@value":"2019-07-09T13:32","@language":"en"}],"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.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/73137?expand=metadata","@language":"en"}],"Extent":[{"@value":"51.0 minutes","@language":"en"}],"FileFormat":[{"@value":"video\/mp4","@language":"en"}],"FullText":[{"@value":"","@language":"en"}],"GeographicLocation":[{"@value":"Banff (Alta.)","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0387498","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Notes":[{"@value":"Author affiliation: MIT","@language":"en"}],"PeerReviewStatus":[{"@value":"Unreviewed","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"Banff International Research Station for Mathematical Innovation and Discovery","@language":"en"}],"Rights":[{"@value":"Attribution-NonCommercial-NoDerivatives 4.0 International","@language":"en"}],"RightsURI":[{"@value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Series":[{"@value":"BIRS Workshop Lecture Videos (Banff, Alta)","@language":"en"}],"Subject":[{"@value":"Mathematics","@language":"en"},{"@value":"Computer Science, Theoretical Computer Science","@language":"en"}],"Title":[{"@value":"Efficient Construction of Rigid Matrices Using an NP Oracle","@language":"en"}],"Type":[{"@value":"Moving Image","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/73137","@language":"en"}],"SortDate":[{"@value":"2019-07-09 AD","@language":"en"}],"@id":"doi:10.14288\/1.0387498"}