TY - ELEC
AU - Josh Alman
PY - 2019
TI - Efficient Construction of Rigid Matrices Using an NP Oracle
LA - eng
M3 - Moving Image
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/48630/items/1.0387498
ER - End of Reference