Non UBC
DSpace
Josh Alman
2020-01-06T09:39:23Z
2019-07-09T13:32
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:
- 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.
Using known connections between matrix rigidity and a number of different areas of complexity theory, we derive several consequences of our constructions, including:
- 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.
- 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.
Joint work with Lijie Chen to appear in FOCS 2019.
https://circle.library.ubc.ca/rest/handle/2429/73137?expand=metadata
51.0 minutes
video/mp4
Author affiliation: MIT
Banff (Alta.)
10.14288/1.0387498
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Graduate
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Computer Science, Theoretical Computer Science
Efficient Construction of Rigid Matrices Using an NP Oracle
Moving Image
http://hdl.handle.net/2429/73137