"Non UBC"@en .
"DSpace"@en .
"Josh Alman"@en .
"2020-01-06T09:39:23Z"@en .
"2019-07-09T13:32"@en .
"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."@en .
"https://circle.library.ubc.ca/rest/handle/2429/73137?expand=metadata"@en .
"51.0 minutes"@en .
"video/mp4"@en .
""@en .
"Author affiliation: MIT"@en .
"Banff (Alta.)"@en .
"10.14288/1.0387498"@en .
"eng"@en .
"Unreviewed"@en .
"Vancouver : University of British Columbia Library"@en .
"Banff International Research Station for Mathematical Innovation and Discovery"@en .
"Attribution-NonCommercial-NoDerivatives 4.0 International"@en .
"http://creativecommons.org/licenses/by-nc-nd/4.0/"@en .
"Graduate"@en .
"BIRS Workshop Lecture Videos (Banff, Alta)"@en .
"Mathematics"@en .
"Computer Science, Theoretical Computer Science"@en .
"Efficient Construction of Rigid Matrices Using an NP Oracle"@en .
"Moving Image"@en .
"http://hdl.handle.net/2429/73137"@en .