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Register a new userSingularity of random integer matrices with large entries
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Added by
Sankeerth Rao Karingula
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We study the singularity probability of random integer matrices. Concretely, the probability that a random $n times n$ matrix, with integer entries chosen uniformly from ${-m,ldots,m}$, is singular. This problem has been well studied in two regimes: large $n$ and constant $m$; or large $m$ and constant $n$. In this paper, we extend previous techniques to handle the regime where both $n,m$ are large. We show that the probability that such a matrix is singular is $m^{-cn}$ for some absolute constant $c>0$. We also provide some connections of our result to coding theory.
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We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Let $xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(xi)$ denote an $ntimes n$ random matrix with entries that are independent copies of $xi$. For $xi$ which is not uniform on its support, we show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= mathbb{P}[text{zero row or column}] + (1+o_n(1))mathbb{P}[text{two equal (up to sign) rows or columns}], end{align*} thereby confirming a folklore conjecture. As special cases, we obtain: (1) For $xi = text{Bernoulli}(p)$ with fixed $p in (0,1/2)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},] which determines the singularity probability to two asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. (2) For $xi = text{Bernoulli}(p)$ with fixed $p in (1/2,1)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.] Previously, only the much weaker upper bound of $(sqrt{p} + o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood. For $xi$ which is uniform on its support: (1) We show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= (1+o_n(1))^{n}mathbb{P}[text{two rows or columns are equal}]. end{align*} (2) Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible part of the unit sphere to the lower tail of the smallest singular value of $M_{n}(xi)$.
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 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 $delta in (0,1)$ such that there is an FNP-machine that, for infinitely many $N$, on input $1^N$ outputs $N times N$ matrices with entries in $mathbb{F}_2$ that are $delta N^2$-far (in Hamming distance) from matrices of rank at most $2^{log N/Omega(log log N)}$. Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed--Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothness-inducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.
We introduce a simple logical inference structure we call a $textsf{spanoid}$ (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry, algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs). One central parameter we study is the $textsf{rank}$ of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework. Another parameter we explore is the $textsf{functional rank}$ of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main questions we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a first step, we develop an entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs.
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