Efficient Construction of Rigid Matrices Using an NP

Efficient Construction of Rigid Matrices Using an NP Oracle Josh Alman Harvard Lijie Chen MIT

Matrix Rigidity •

Matrix Rigidity • Both are notoriously hard questions open for decades

What’s Known? Technique Running Time

What’s Known? Technique Running Time Combinatorial Argument (“untouched minor argument”) P

What’s Known? Technique Running Time Combinatorial Argument (“untouched minor argument”) P Random Matrix is Rigid w. h. p.

What’s Known? Technique Running Time Combinatorial Argument (“untouched minor argument”) P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E

What’s Known? Technique Running Time Combinatorial Argument (“untouched minor argument”) P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E

What’s Known? Technique Running Time Combinatorial Argument (“untouched minor argument”) P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E Our New Result! PNP

What’s Known? Technique Combinatorial Argument (“untouched minor argument”) Running Time P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E Our New Result! PNP

What’s Known? Technique Combinatorial Argument (“untouched minor argument”) Running Time P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E Our New Result! PNP

What’s Known? Technique Combinatorial Argument (“untouched minor argument”) Running Time P Random Matrix is Rigid w. h. p. Random 0/1 Toeplitz Matrix is Rigid [Goldreich, Tal ‘ 16] E Our New Result! PNP

What’s Known? Running Time Previous works: mostly combinational or algebraic Combinatorial Argument P Our work: builds on complexity-theoretical ideas and uses PCP! (“untouched minor argument”) Random Matrix is Rigid w. h. p. We take a very Random 0/1 Toeplitz Matrix is Rigid E different approach! [Goldreich, Tal ‘ 16] Our New Result! PNP Technique

Application: Communication Complexity PHcc = Communication Complexity analogue of Polynomial Hierarchy •

Application: Depth-2 Circuit Lower Bound •

Proof Overview Constructing rigid matrices with Algorithms from fine-grained complexity and PCPs

Think about Low-Rank Matrices as a Circuit Class • A B

Williams’ Algorithmic Approach to Circuit LBs Non-trivial Circuit-Analysis Algorithms [Williams ‘ 14] Circuit Lower Bounds! Idea: Use [Williams ‘ 14] approach to turn the above algorithm into lower bounds!

First Attempt do a binary search

First Attempt

First Attempt

Bootstrapping Low-Rank Approximations

Summary •

Thanks! Any Questions?