Rank deficiency of random matrices
Let $M_n$ be a random $n\times n$ matrix with i.i.d. $\text{Bernoulli}(1/2)$ entries. We show that for fixed $k\ge 1$, \[\lim_{n\to \infty}\frac{1}{n}\log_2\mathbb{P}[\text{corank }M_n\ge k] = -k.\]
math.PR↗
arXiv subjects
Vishesh Jain
Publications and source records attributed to Vishesh Jain.
On the sampling Lovász Local Lemma for atomic constraint satisfaction problems
We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probability of violation of any constraint and let $Δ$ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lovász local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide sampling algorithms for the uniform distribution on: (1) $q$-colorings of $k$-uniform hypergraphs with $Δ\lesssim q^{(k-4)/3 + o_{q}(1)}.$ The exponent $1/3$ improves the previously best-known $1/7$ in the case $q, Δ= O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $1/9$ in the general case [Feng, He, Yin; STOC 2021]. (2) Satisfying assignments of Boolean $k$-CNF formulas with $Δ\lesssim 2^{k/5.741}.$ The constant $5.741$ in the exponent improves the previously best-known $7$ in the case $k = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $13$ in the general case [Feng, He, Yin; STOC 2021]. (3) Satisfying assignments of general atomic constraint satisfaction problems with $p\cdot Δ^{7.043} \lesssim 1.$ The constant $7.043$ improves upon the previously best-known constant of $350$ [Feng, He, Yin; STOC 2021]. At the heart of our analysis is a novel information-percolation type argument for showing the rapid mixing of the Glauber dynamics for a carefully constructed projection of the uniform distribution on satisfying assignments. Notably, there is no natural partial order on the space, and we believe that the techniques developed for the analysis may be of independent interest.
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Optimal and algorithmic norm regularization of random matrices
Let $A$ be an $n\times n$ random matrix whose entries are i.i.d. with mean $0$ and variance $1$. We present a deterministic polynomial time algorithm which, with probability at least $1-2\exp(-Ω(εn))$ in the choice of $A$, finds an $εn \times εn$ sub-matrix such that zeroing it out results in $\widetilde{A}$ with \[\|\widetilde{A}\| = O\left(\sqrt{n/ε}\right).\] Our result is optimal up to a constant factor and improves previous results of Rebrova and Vershynin, and Rebrova. We also prove an analogous result for $A$ a symmetric $n\times n$ random matrix whose upper-diagonal entries are i.i.d. with mean $0$ and variance $1$.
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Towards the sampling Lovász Local Lemma
Let $Φ= (V, \mathcal{C})$ be a constraint satisfaction problem on variables $v_1,\dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that each constraint shares variables with at most $Δ$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that \[C\cdot q^{3}\cdot k \cdot p \cdot Δ^{7} < 1, \quad \text{where }C \text{ is an absolute constant.}\] Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$-CNF formulas, the term $Δ^{7}$ improves the previously best known $Δ^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $Δ^{13}$ for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $Δ^{7}$ improves the previously best known $Δ^{14}$ for deterministic algorithms [Guo et al., SICOMP, 2019] and $Δ^{9}$ for randomized algorithms [Feng et al., arXiv, 2020].
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On the smallest singular value of symmetric random matrices
We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $ξ$ with mean $0$ and variance $1$, \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O_ξ(ε^{1/8} + \exp(-Ω_ξ(n^{1/2}))) \quad \text{for all } ε\ge 0.\] This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant $c$, and $1/8$ replaced by $(1/8) + η$ (with implicit constants also depending on $η> 0$). Furthermore, when $ξ$ is a Rademacher random variable, we prove that \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O(ε^{1/8} + \exp(-Ω((\log{n})^{1/4}n^{1/2}))) \quad \text{for all } ε\ge 0.\] The special case $ε= 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(-Ω(n^{1/2}))).$ The main innovation in our work are new notions of arithmetic structure -- the Median Regularized Least Common Denominator and the Median Threshold, which we believe should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.
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On the smoothed analysis of the smallest singular value with discrete noise
Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and variance $1$. We make two contributions to the study of $s_n(A+M)$, the smallest singular value of $A+M$. (1) We show that for all $ε\geq 0$, $$\mathbb{P}[s_n(A + M) \leq ε] = O(ε\sqrt{n}) + 2e^{-Ω(n)},$$ provided only that $A$ has $Ω(n)$ singular values which are $O(\sqrt{n})$. This extends a well-known result of Rudelson and Vershynin, which requires all singular values of $A$ to be $O(\sqrt{n})$. (2) We show that any bound of the form $$\sup_{\|{A}\|\leq n^{C_1}}\mathbb{P}[s_n(A+M)\leq n^{-C_3}] \leq n^{-C_2}$$ must have $C_3 = Ω(C_1 \sqrt{C_2})$. This complements a result of Tao and Vu, who proved such a bound with $C_3 = O(C_1C_2 + C_1 + 1)$, and counters their speculation of possibly taking $C_3 = O(C_1 + C_2)$.
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The smallest singular value of dense random regular digraphs
Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geqλn$. We show that for any $κ\geq 0$, \[\mathbb{P}[s_n(A)\leqκ]\leq C_λκ\sqrt{n}+2e^{-c_λn}.\] Up to the constants $C_λ, c_λ> 0$, our bound matches optimal bounds for $n\times n$ random matrices, each of whose entries is an i.i.d $\text{Ber}(d/n)$ random variable. The special case $κ= 0$ of our result confirms a conjecture of Cook regarding the probability of singularity of dense random regular digraphs.
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Fast and memory-optimal dimension reduction using Kac's walk
In this work, we analyze dimension reduction algorithms based on the Kac walk and discrete variants. (1) For $n$ points in $\mathbb{R}^{d}$, we design an optimal Johnson-Lindenstrauss (JL) transform based on the Kac walk which can be applied to any vector in time $O(d\log{d})$ for essentially the same restriction on $n$ as in the best-known transforms due to Ailon and Liberty [SODA, 2008], and Bamberger and Krahmer [arXiv, 2017]. Our algorithm is memory-optimal, and outperforms existing algorithms in regimes when $n$ is sufficiently large and the distortion parameter is sufficiently small. In particular, this confirms a conjecture of Ailon and Chazelle [STOC, 2006] in a stronger form. (2) The same construction gives a simple transform with optimal Restricted Isometry Property (RIP) which can be applied in time $O(d\log{d})$ for essentially the same range of sparsity as in the best-known such transform due to Ailon and Rauhut [Discrete Comput. Geom., 2014]. (3) We show that by fixing the angle in the Kac walk to be $π/4$ throughout, one obtains optimal JL and RIP transforms with almost the same running time, thereby confirming -- up to a $\log\log{d}$ factor -- a conjecture of Avron, Maymounkov, and Toledo [SIAM J. Sci. Comput., 2010]. Our moment-based analysis of this modification of the Kac walk may also be of independent interest.
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Perfectly Sampling $k\geq (8/3 +o(1))Δ$-Colorings in Graphs
We present a randomized algorithm which takes as input an undirected graph $G$ on $n$ vertices with maximum degree $Δ$, and a number of colors $k \geq (8/3 + o_Δ(1))Δ$, and returns -- in expected time $\tilde{O}(nΔ^{2}\log{k})$ -- a proper $k$-coloring of $G$ distributed perfectly uniformly on the set of all proper $k$-colorings of $G$. Notably, our sampler breaks the barrier at $k = 3Δ$ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methods to relax the restriction on $k$ to $k \geq (8/3 - ε_0)Δ$ for an absolute constant $ε_0 > 0$. As in the work of Bhandari and Chakraborty, and the pioneering work of Huber [STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson, Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC 1998; Häggström&Nelander, Scand. J. Statist., 1999]. Our innovations include a novel bounding chain routine inspired by Jerrum's analysis of the Glauber dynamics [Random Struct. Algorithms, 1995], as well as a preconditioning routine for bounding chains which uses the algorithmic Lovász Local Lemma [Moser&Tardos, J.ACM, 2010].
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Universality and least singular values of random matrix products: a simplified approach
In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Koppel, O'Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality . In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from $0$ and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed. A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Koppel, O'Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.
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A note on the universality of ESDs of inhomogeneous random matrices
In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption. Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at $0$ and infinity, typically have smallest singular value $Ω(n^{-1/2})$. The rate $n^{-1/2}$ is sharp, and closes a gap in the literature. Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov, and Vershynin on inhomogeneous real random matrices. The new ingredient is a couple of anti-concentration inequalities for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.
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The strong circular law: a combinatorial view
Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is an independent copy of a centered complex random variable $z$ with finite non-zero variance $σ^{2}$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(σ\sqrt{n})$ converges to the uniform distribution on the unit disc in $\mathbb{C}$. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix $xI - N_{n}/(σ\sqrt{n})$ (where $x\in \mathbb{C}$ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using tools from additive combinatorics or any net arguments), we show that for any fixed matrix $M$ with operator norm at most $n^{0.51}$ and for all $η\geq 0$, $$\Pr\left(s_n(M+N_n) \leq η\right) \lesssim n^{C}η+ \exp(-n^{c}),$$ where $s_n(M+N_n)$ is the least singular value of $M+N_n$ and $C,c$ are absolute constants. Our result is optimal up to the constants $C,c$ and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. During the course of our proof, we extend the solution of the counting problem in inverse Littlewood-Offord theory, recently isolated by the author along with Ferber, Luh, and Samotij, from Rademacher variables to general complex random variables. This significantly improves on estimates for this problem obtained using the optimal inverse Littlewood-Offord theorem of Nguyen and Vu, and may be of independent interest.
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On the real Davies' conjecture
We show that every matrix $A \in \mathbb{R}^{n\times n}$ is at least $δ$$\|A\|$-close to a real matrix $A+E \in \mathbb{R}^{n\times n}$ whose eigenvectors have condition number at most $\tilde{O}_{n}(δ^{-1})$. In fact, we prove that, with high probability, taking $E$ to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies, and of Banks, Kulkarni, Mukherjee, and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices. Along the way, we also prove non-asymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.
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The probability of selecting $k$ edge-disjoint Hamilton cycles in the complete graph
Let $H_1,\dots,H_k$ be Hamilton cycles in $K_n$, chosen independently and uniformly at random. We show, for $k = o(n^{1/100})$, that the probability of $H_1,\dots,H_k$ being edge-disjoint is $(1+o(1))e^{-2\binom{k}{2}}$. This extends a corresponding estimate obtained by Robbins in the case $k=2$.
math.CO↗
Singularity of random symmetric matrices -- a combinatorial approach to improved bounds
Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_n$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_n$, due to Vershynin (2011), is $2^{-n^c}$, for some unspecified small constant $c > 0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_n$ is at most $2^{-n^{1/4}\sqrt{\log{n}}/1000}$ for all sufficiently large $n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.
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Number of 1-factorizations of regular high-degree graphs
A $1$-factor in an $n$-vertex graph $G$ is a collection of $\frac{n}{2}$ vertex-disjoint edges and a $1$-factorization of $G$ is a partition of its edges into edge-disjoint $1$-factors. Clearly, a $1$-factorization of $G$ cannot exist unless $n$ is even and $G$ is regular (that is, all vertices are of the same degree). The problem of finding $1$-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has a $1$-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, Kühn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950s, which says that every $d$-regular graph on $n$ vertices contains a $1$-factorization, provided that $n$ is even and $d\geq 2\lceil \frac{n}{4}\rceil-1$. In this paper we address the natural question of estimating $F(n,d)$, the number of $1$-factorizations in $d$-regular graphs on an even number of vertices, provided that $d\geq \frac{n}{2}+\varepsilon n$. Improving upon a recent result of Ferber and Jain, which itself improved upon a result of Cameron from the 1970s, we show that $F(n,d)\geq \left((1+o(1))\frac{d}{e^2}\right)^{nd/2}$, which is asymptotically best possible.
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Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation
The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.
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Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices
An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $η\geq 0$, $\Pr(s_n(M_n) \leq η) \lesssim n^{C}η+ \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a `truly combinatorial' setting.
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