arXiv subjects

Thuy-Duong Vuong

Publications and source records attributed to Thuy-Duong Vuong.

26 records · Page 2Linked to original sources

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 $(V,E,d')$.

math.PR↗

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 $n$.

cs.DS↗

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.

cs.DS↗

arXiv2021

An Extension of Plücker Relations with Applications to Subdeterminant Maximization

Given a matrix $A$ and $k\geq 0$, we study the problem of finding the $k\times k$ submatrix of $A$ with the maximum determinant in absolute value. This problem is motivated by the question of computing the determinant-based lower bound of [LSV86] on hereditary discrepancy, which was later shown to be an approximate upper bound as well [Mat13]. The special case where $k$ coincides with one of the dimensions of $A$ has been extensively studied. [Nik15] gave a $2^{O(k)}$-approximation algorithm for this special case, matching known lower bounds; he also raised as an open problem the question of designing approximation algorithms for the general case. We make progress towards answering this question by giving the first efficient approximation algorithm for general $k\times k$ subdeterminant maximization with an approximation ratio that depends only on $k$. Our algorithm finds a $k^{O(k)}$-approximate solution by performing a simple local search. Our main technical contribution, enabling the analysis of the approximation ratio, is an extension of Plücker relations for the Grassmannian, which may be of independent interest; Plücker relations are quadratic polynomial equations involving the set of $k\times k$ subdeterminants of a $k\times n$ matrix. We find an extension of these relations to $k\times k$ subdeterminants of general $m\times n$ matrices.

cs.DS↗

arXiv2021

From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization

We show a connection between sampling and optimization on discrete domains. For a family of distributions $μ$ defined on size $k$ subsets of a ground set of elements that is closed under external fields, we show that rapid mixing of natural local random walks implies the existence of simple approximation algorithms to find $\max μ(\cdot)$. More precisely we show that if (multi-step) down-up random walks have spectral gap at least inverse polynomially large in $k$, then (multi-step) local search can find $\max μ(\cdot)$ within a factor of $k^{O(k)}$. As the main application of our result, we show a simple nearly-optimal $k^{O(k)}$-factor approximation algorithm for MAP inference on nonsymmetric DPPs. This is the first nontrivial multiplicative approximation for finding the largest size $k$ principal minor of a square (not-necessarily-symmetric) matrix $L$ with $L+L^\intercal\succeq 0$. We establish the connection between sampling and optimization by showing that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks. We further connect exchange inequalities with composable core-sets for optimization, generalizing recent results on composable core-sets for DPP maximization to arbitrary distributions that satisfy either the strongly Rayleigh property or that have a log-concave generating polynomial.

cs.LG↗

arXiv2021

Domain Sparsification of Discrete Distributions using Entropic Independence

We present a framework for speeding up the time it takes to sample from discrete distributions $μ$ defined over subsets of size $k$ of a ground set of $n$ elements, in the regime $k\ll n$. We show that having estimates of marginals $\mathbb{P}_{S\sim μ}[i\in S]$, the task of sampling from $μ$ can be reduced to sampling from distributions $ν$ supported on size $k$ subsets of a ground set of only $n^{1-α}\cdot \operatorname{poly}(k)$ elements. Here, $1/α\in [1, k]$ is the parameter of entropic independence for $μ$. Further, the sparsified distributions $ν$ are obtained by applying a sparse (mostly $0$) external field to $μ$, an operation that often retains algorithmic tractability of sampling from $ν$. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of $μ$, and in return reduce the amortized cost needed to produce many samples from the distribution $μ$, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where $α=Ω(1)$, our result reduces the domain size, and as a corollary, the cost-per-sample, by a $\operatorname{poly}(n)$ factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to $α=1$). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over $O(1/k)$ relative error established in prior work.

cs.DS↗

arXiv2021

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.

cs.DS↗

arXiv2016

Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its brane tiling, which arise from the physics literature, in terms of toric mutations on its corresponding cluster. Specifically, we give explicit formulas for all cluster variables generated by toric mutation sequences. Moreover, for each such variable, we associate a subgraph of the dP$_2$ brane tiling to it such that its weight matches the variable.

math.CO↗