
The rank of sparse random matrices
Generalising prior work on the rank of random matrices over finite field...
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Recent Progress on Matrix Rigidity – A Survey
The concept of matrix rigidity was introduced by Valiant(independently b...
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Distance Estimation Between Unknown Matrices Using Sublinear Projections on Hamming Cube
Using geometric techniques like projection and dimensionality reduction,...
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Fourier and Circulant Matrices are Not Rigid
The concept of matrix rigidity was first introduced by Valiant in [Val77...
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The rank of random matrices over finite fields
We determine the rank of a random matrix A over a finite field with pres...
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On the structure of matrices avoiding intervalminor patterns
We study the structure of 01matrices avoiding a pattern P as an interva...
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Matrix rigidity and the CrootLevPach lemma
Matrix rigidity is a notion put forth by Valiant as a means for proving ...
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Rigid Matrices From Rectangular PCPs
We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the *column*. We construct PCPs that are efficient, short, smooth and (almost)rectangular. As a key application, we show that proofs for hard languages in NTIME(2^n), when viewed as matrices, are rigid infinitely often. This strengthens and considerably simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem:  There is a constant δ∈ (0,1) such that there is an FNPmachine that, for infinitely many N, on input 1^N outputs N × N matrices with entries in 𝔽_2 that are δ N^2far (in Hamming distance) from matrices of rank at most 2^log N/Ω(loglog N). Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed–Mullerbased outer PCP of BenSasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothnessinducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.
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