[{"key":"dc.contributor.author","value":"Josh Alman","language":null},{"key":"dc.coverage.spatial","value":"Banff (Alta.)","language":null},{"key":"dc.date.accessioned","value":"2020-01-06T06:00:52Z","language":null},{"key":"dc.date.available","value":"2020-01-06T09:39:23Z","language":null},{"key":"dc.date.issued","value":"2019-07-09T13:32","language":null},{"key":"dc.identifier.other","value":"BIRS-VIDEO-201907091332-Alman","language":null},{"key":"dc.identifier.other","value":"BIRS-VIDEO-19w5088-33621","language":null},{"key":"dc.identifier.uri","value":"http:\/\/hdl.handle.net\/2429\/73137","language":null},{"key":"dc.description.abstract","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":null},{"key":"dc.format.extent","value":"51.0 minutes","language":null},{"key":"dc.format.mimetype","value":"video\/mp4","language":null},{"key":"dc.language.iso","value":"eng","language":null},{"key":"dc.publisher","value":"Banff International Research Station for Mathematical Innovation and Discovery","language":null},{"key":"dc.relation","value":"19w5088: Algebraic Techniques in Computational Complexity","language":null},{"key":"dc.relation.ispartofseries","value":"BIRS Workshop Lecture Videos (Banff, Alta)","language":null},{"key":"dc.rights","value":"Attribution-NonCommercial-NoDerivatives 4.0 International","language":null},{"key":"dc.rights.uri","value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","language":null},{"key":"dc.subject","value":"Mathematics","language":null},{"key":"dc.subject","value":"Computer Science, Theoretical Computer Science","language":null},{"key":"dc.title","value":"Efficient Construction of Rigid Matrices Using an NP Oracle","language":null},{"key":"dc.type","value":"Moving Image","language":null},{"key":"dc.description.affiliation","value":"Non UBC","language":null},{"key":"dc.description.reviewstatus","value":"Unreviewed","language":null},{"key":"dc.description.notes","value":"Author affiliation: MIT","language":null},{"key":"dc.description.scholarlevel","value":"Graduate","language":null},{"key":"dc.date.updated","value":"2020-01-06T06:00:52Z","language":null}]