arXiv subjects

Vishesh Jain

Publications and source records attributed to Vishesh Jain.

At least 37 records · Page 2Linked to original sources

arXiv2024

Rapid mixing of the down-up walk on matchings of a fixed size

Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $δ> 0$ and for any $1 \leq k \leq (1-δ)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes in time polynomial in $n$. Previously, polynomial mixing was not known even for graphs with maximum degree $Δ$, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time $O_{Δ,δ}(n\log{n})$. In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.

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arXiv2024

An explicit economical additive basis

We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots\}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon > 0$, \[\lim_{N\to \infty}\frac{\big|\big\{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big\}\big|}{N^{\varepsilon}} = 0.\] This answers a question of Erdős.

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arXiv2023

Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

We study Glauber dynamics for sampling from discrete distributions $μ$ on the hypercube $\{\pm 1\}^n$. Recently, techniques based on spectral independence have successfully yielded optimal $O(n)$ relaxation times for a host of different distributions $μ$. We show that spectral independence is universal: a relaxation time of $O(n)$ implies spectral independence. We then study a notion of tractability for $μ$, defined in terms of smoothness of the multilinear extension of its Hamiltonian -- $\log μ$ -- over $[-1,+1]^n$. We show that Glauber dynamics has relaxation time $O(n)$ for such $μ$, and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal $\widetilde{O}(n)$ mixing time for random Hamiltonians, i.e. the classically studied mixed $p$-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.

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arXiv2023

Optimal mixing of the down-up walk on independent sets of a given size

Let $G$ be a graph on $n$ vertices of maximum degree $Δ$. We show that, for any $δ> 0$, the down-up walk on independent sets of size $k \leq (1-δ)α_c(Δ)n$ mixes in time $O_{Δ,δ}(k\log{n})$, thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, $α_{c}(Δ)n$ is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on $n$ vertices of maximum degree $Δ$. Our mixing time has optimal dependence on $k,n$ for the entire range of $k$; previously, even polynomial mixing was not known. In fact, for $k = Ω_Δ(n)$ in this range, we establish a log-Sobolev inequality with optimal constant $Ω_{Δ,δ}(1/n)$. At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting $\ell_\infty$-independence from a suitable distribution on the discrete cube -- in this case, the hard-core model -- to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.

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arXiv2022

Optimal thresholds for Latin squares, Steiner Triple Systems, and edge colorings

We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $Θ(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list edge-colorings of the complete (bipartite) graph with random lists. Our results answer several related questions of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus.

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arXiv2022

Spencer's theorem in nearly input-sparsity time

A celebrated theorem of Spencer states that for every set system $S_1,\dots, S_m \subseteq [n]$, there is a coloring of the ground set with $\{\pm 1\}$ with discrepancy $O(\sqrt{n\log(m/n+2)})$. We provide an algorithm to find such a coloring in near input-sparsity time $\tilde{O}(n+\sum_{i=1}^{m}|S_i|)$. A key ingredient in our work, which may be of independent interest, is a novel width reduction technique for solving linear programs, not of covering/packing type, in near input-sparsity time using the multiplicative weights update method.

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arXiv2022

Optimal minimization of the covariance loss

Let $X$ be a random vector valued in $\mathbb{R}^{m}$ such that $\|X\|_{2} \le 1$ almost surely. For every $k\ge 3$, we show that there exists a sigma algebra $\mathcal{F}$ generated by a partition of $\mathbb{R}^{m}$ into $k$ sets such that \[\|\operatorname{Cov}(X) - \operatorname{Cov}(\mathbb{E}[X\mid\mathcal{F}]) \|_{\mathrm{F}} \lesssim \frac{1}{\sqrt{\log{k}}}.\] This is optimal up to the implicit constant and improves on a previous bound due to Boedihardjo, Strohmer, and Vershynin. Our proof provides an efficient algorithm for constructing $\mathcal{F}$ and leads to improved accuracy guarantees for $k$-anonymous or differentially private synthetic data. We also establish a connection between the above problem of minimizing the covariance loss and the pinning lemma from statistical physics, providing an alternate (and much simpler) algorithmic proof in the important case when $X \in \{\pm 1\}^m/\sqrt{m}$ almost surely.

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arXiv2022

Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Let $G = (V,E)$ be an undirected graph with maximum degree $Δ$ and vertex conductance $Ψ^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $γ^*(P)$ satisfies \[Ψ^*(G)^{2}/\logΔ\lesssim γ^*(P) \lesssim Ψ^*(G).\] Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with $\logΔ$ replaced by $\log|V|$. In order to obtain our result, we show how to embed a negative-type semi-metric $d$ defined on $V$ into a negative-type semi-metric $d'$ supported in $\mathbb{R}^{O(\logΔ)}$, such that the (fractional) matching number of the weighted graph $(V,E,d)$ is approximately equal to that of $(V,E,d')$.

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arXiv2021

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $μ$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. To derive our results, we relate entropic independence to properties of polynomials: $μ$ is entropically independent exactly when a transformed version of the generating polynomial of $μ$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions. As our flagship application, we establish the tight mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$.

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arXiv2021

Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities

We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for the Markov operator, and are notoriously difficult to establish. However, infinitesimally close to stationarity, entropy contraction becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which is significantly easier to establish through, e.g., spectral analysis. Motivated by this observation, we study restricted modified log-Sobolev inequalities that guarantee entropy contraction not for all starting distributions, but for those in a large neighborhood of the stationary distribution. We show how to sample from the hardcore and Ising models on $n$-node graphs that have a constant $δ$ relative gap to the tree-uniqueness threshold, in nearly-linear time $\widetilde O_δ(n)$. Notably, our bound does not depend on the maximum degree $Δ$, and is therefore optimal even for high-degree graphs. This improves on prior mixing time bounds of $\widetilde O_{δ, Δ}(n)$ and $\widetilde O_δ(n^2)$, established via (non-restricted) modified log-Sobolev and Poincare inequalities respectively. We further show that optimal concentration inequalities can still be achieved from the restricted form of modified log-Sobolev inequalities. To establish restricted entropy contraction, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to the setting of distributions that are spectrally independent under a restricted set of external fields. We also develop an orthogonal trick that might be of independent interest: utilizing Bernoulli factories we show how to implement Glauber dynamics updates on high-degree graphs in $O(1)$ time, assuming standard adjacency array representation of the graph.

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arXiv2021

Quantitative invertibility of random matrices: a combinatorial perspective

We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix. This setting has previously been considered in a series of influential works by Tao and Vu, most notably in connection with the strong circular law, and the smoothed analysis of the condition number, and our results improve upon theirs in two ways: (i) We are able to handle $\|M\| = O(\exp(n^{c}))$, whereas the results of Tao and Vu are applicable only for $M = O(\text{poly(n)})$. (ii) Even for $M = O(\text{poly(n)})$, we are able to extract more refined information -- for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(\exp(-n^{c}))$, whereas even in the case when $ξ$ is a Bernoulli random variable, the results of Tao and Vu only give a bound of the form $O_{C}(n^{-C})$ for any constant $C>0$. As opposed to all previous works obtaining such bounds with error rate better than $n^{-1}$, our proof makes no use either of the inverse Littlewood--Offord theorems, or of any sophisticated net constructions. Instead, we show how to reduce the problem from the (complex) sphere to (Gaussian) integer vectors, where it is solved directly by utilizing and extending a combinatorial approach to the singularity problem for random discrete matrices, recently developed by Ferber, Luh, Samotij, and the author.

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arXiv2021

Approximate counting and sampling via local central limit theorems

We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $Δ$ on $n$ vertices, for all $k \le (1-δ)m^*(G)$, where $δ>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an FPTAS for the number of independent sets of size $k \le (1-δ) α_c(Δ) n$, where $α_c(Δ)$ is the NP-hardness threshold for this problem. We also provide quasi-linear time randomized algorithms to approximately sample from the uniform distribution on matchings of size $k \leq (1-δ)m^*(G)$ and independent sets of size $k \leq (1-δ)α_c(Δ)n$. Our results are based on a new framework for exploiting local central limit theorems as an algorithmic tool. We use a combination of Fourier inversion, probabilistic estimates, and the deterministic approximation of partition functions at complex activities to extract approximations of the coefficients of the partition function. For our results for independent sets, we prove a new local central limit theorem for the hard-core model that applies to all fugacities below $λ_c(Δ)$, the uniqueness threshold on the infinite $Δ$-regular tree.

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arXiv2021

Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $τ\in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-τ}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting.

math.PR↗

arXiv2021

Anticoncentration versus the number of subset sums

Let $\vec{w} = (w_1,\dots, w_n) \in \mathbb{R}^{n}$. We show that for any $n^{-2}\leε\le 1$, if \[\#\{\vecξ \in \{0,1\}^{n}: \langle \vecξ, \vec{w} \rangle = τ\} \ge 2^{-εn}\cdot 2^{n}\] for some $τ\in \mathbb{R}$, then \[\#\{\langle \vecξ, \vec{w} \rangle : \vecξ \in \{0,1\}^{n}\} \le 2^{O(\sqrtεn)}.\] This exponentially improves the $ε$ dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Węgrzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.

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arXiv2021

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$-matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$. Previously, such estimates were available only for $p = o(n^{1/8})$. At the heart of our proof is a way to combine multiple inverse Littlewood--Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + Ω(1/p^2)$. Previously, inverse Littlewood--Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$.

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arXiv2021

Sharp invertibility of random Bernoulli matrices

Let $p \in (0,1/2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t \ge 0$, \[\mathbb{P}[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n} + C_p (1-p-ε_p)^{n},\] where $s_n(B_n(p))$ denotes the least singular value of $B_n(p)$ and $C_p, ε_p > 0$ are constants depending only on $p$. In particular, \[\mathbb{P}[B_{n}(p) \text{ is singular}] = 2n(1-p)^{n} + C_{p}(1-p-ε_p)^{n},\] which confirms a conjecture of Litvak and Tikhomirov. We also confirm a conjecture of Nguyen by showing that if $Q_{n}$ is an $n\times n$ random matrix with independent rows that are uniformly distributed on the central slice of $\{0,1\}^{n}$, then \[\mathbb{P}[Q_{n} \text{ is singular}] = (1/2 + o_n(1))^{n}.\] This provides, for the first time, a sharp determination of the logarithm of the probability of singularity in any natural model of random discrete matrices with dependent entries.

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arXiv2021

Singularity of discrete random matrices

Let $ξ$ be a non-constant real-valued random variable with finite support, and let $M_{n}(ξ)$ denote an $n\times n$ random matrix with entries that are independent copies of $ξ$. For $ξ$ which is not uniform on its support, we show that \begin{align*} \mathbb{P}[M_{n}(ξ)\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 $ξ= \text{Bernoulli}(p)$ with fixed $p \in (0,1/2)$, \[\mathbb{P}[M_{n}(ξ)\text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},\] 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 $ξ= \text{Bernoulli}(p)$ with fixed $p \in (1/2,1)$, \[\mathbb{P}[M_{n}(ξ)\text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.\] 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 $ξ$ which is uniform on its support: (1) We show that \begin{align*} \mathbb{P}[M_{n}(ξ)\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 $M_{n}(ξ)$.

math.PR↗

arXiv2021

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent $q$-spin system on a graph $G = (V,E)$ with maximum degree $Δ$. Notably, for several interesting examples, our bound covers the entire regime of $Δ$ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph $G = (V,E)$ with maximum degree $Δ\geq 3$, the Glauber dynamics for the uniform distribution on proper $k$-colorings with $k \geq (1.763\dots + δ)Δ$ colors has spectral gap $\tildeΩ_δ(|V|^{-1})$. Previously, such a result was known either if the girth of $G$ is at least $5$ [Dyer et.~al, FOCS 2004], or under restrictions on $Δ$ [Chen et.~al, STOC 2021; Hayes-Vigoda, FOCS 2003]. (2) We show that for a regular graph $G = (V,E)$ with degree $Δ\geq 3$ and girth at least $6$, and for any $\varepsilon, δ> 0$, the partition function of the hardcore model with fugacity $λ\leq (1-δ)λ_{c}(Δ)$ may be approximated within a $(1+\varepsilon)$-multiplicative factor in time $\tilde{O}_δ(n^{2}\varepsilon^{-2})$. Previously, such a result was known if the girth is at least $7$ [Efthymiou et.~al, SICOMP 2019]. (3) We show for the binomial random graph $G(n,d/n)$ with $d = O(1)$, with high probability, an approximately uniformly random matching may be sampled in time $O_{d}(n^{2+o(1)})$. This improves the corresponding running time of $\tilde{O}_{d}(n^{3})$ due to [Jerrum-Sinclair, SICOMP 1989; Jerrum, 2003].

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