arXiv subjects

Amit Rajaraman

Publications and source records attributed to Amit Rajaraman.

9 recordsLinked to original sources

arXiv2026

A Simple Algorithm for Best Separable State

We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-\epsilon$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.

cs.DS↗

arXiv2025

Markov Chains Approximate Message Passing

Markov chain Monte Carlo algorithms have long been observed to obtain near-optimal performance in various Bayesian inference settings. However, developing a supporting theory that makes these studies rigorous has proved challenging. In this paper, we study the classical spiked Wigner inference problem, where one aims to recover a planted Boolean spike from a noisy matrix measurement. We relate the recovery performance of Glauber dynamics on the annealed posterior to the performance of Approximate Message Passing (AMP), which is known to achieve Bayes-optimal performance. Our main results rely on the analysis of an auxiliary Markov chain called restricted Gaussian dynamics (RGD). Concretely, we establish the following results: 1. RGD can be reduced to an effective one-dimensional recursion which mirrors the evolution of the AMP iterates. 2. From a warm start, RGD rapidly converges to a fixed point in correlation space, which recovers Bayes-optimal performance when run on the posterior. 3. Conditioned on widely believed mixing results for the SK model, we recover the phase transition for non-trivial inference.

cs.DS↗

arXiv2025

Faster MAX-CUT on Bounded Threshold Rank Graphs

We design new algorithms for approximating 2CSPs on graphs with bounded threshold rank, that is, whose normalized adjacency matrix has few eigenvalues larger than $\varepsilon$, smaller than $-\varepsilon$, or both. Unlike on worst-case graphs, 2CSPs on bounded threshold rank graphs can be $(1+O(\varepsilon))$-approximated efficiently. Prior approximation algorithms for this problem run in time exponential in the threshold rank and $1/\varepsilon$. Our algorithm has running time which is polynomial in $1/\varepsilon$ and exponential in the threshold rank of the label-extended graph, and near-linear in the input size. As a consequence, we obtain the first $(1+O(\varepsilon))$ approximation for MAX-CUT on bounded threshold rank graphs running in $\mathrm{poly}(1/\varepsilon)$ time. We also improve the state-of-the-art running time for 2CSPs on bounded threshold-rank graphs from polynomial in input size to near-linear via a new comparison inequality between the threshold rank of the label-extended graph and base graph. Our algorithm is a simple yet novel combination of subspace enumeration and semidefinite programming.

cs.DS↗

arXiv2025

Eigenvalue Bounds for Random Matrices via Zerofreeness

We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our techniques are entirely non-asymptotic, and we instantiate it in three settings: (i) The spectral radius of non-asymptotic Girko matrices -- these are asymmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$ and $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$. (ii) The spectral radius of non-asymptotic Wigner matrices -- these are symmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries above the diagonal are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$, $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$, and $\mathbb{E} |\mathbf{M}_{ij}^4| \le \frac{1}{n}$. (iii) The second eigenvalue of the adjacency matrix of a random $d$-regular graph on $n$ vertices, as drawn from the configuration model. In all three settings, we obtain constant-probability eigenvalue bounds that are tight up to a constant. Applied to specific random matrix ensembles, we recover classic bounds for Wigner matrices, as well as results of Bordenave--Chafa\"{i}--Garc\'{i}a-Zelada, Bordenave--Lelarge--Massouli\'{e}, and Friedman, up to constants.

math.PR↗

arXiv2025

The Fundamental Limits of Recovering Planted Subgraphs

Given an arbitrary subgraph $H=H_n$ and $p=p_n \in (0,1)$, the planted subgraph model is defined as follows. A statistician observes the union a random copy $H^*$ of $H$, together with random noise in the form of an instance of an Erdos-Renyi graph $G(n,p)$. Their goal is to recover the planted $H^*$ from the observed graph. Our focus in this work is to understand the minimum mean squared error (MMSE) for sufficiently large $n$. A recent paper [MNSSZ23] characterizes the graphs for which the limiting MMSE curve undergoes a sharp phase transition from $0$ to $1$ as $p$ increases, a behavior known as the all-or-nothing phenomenon, up to a mild density assumption on $H$. In this paper, we provide a formula for the limiting MMSE curve for any graph $H=H_n$, up to the same mild density assumption. This curve is expressed in terms of a variational formula over pairs of subgraphs of $H$, and is inspired by the celebrated subgraph expectation thresholds from the probabilistic combinatorics literature [KK07]. Furthermore, we give a polynomial-time description of the optimizers of this variational problem. This allows one to efficiently approximately compute the MMSE curve for any dense graph $H$ when $n$ is large enough. The proof relies on a novel graph decomposition of $H$ as well as a new minimax theorem which may be of independent interest. Our results generalize to the setting of minimax rates of recovering arbitrary monotone boolean properties planted in random noise, where the statistician observes the union of a planted minimal element $A \subseteq [N]$ of a monotone property and a random $Ber(p)^{\otimes N}$ vector. In this setting, we provide a variational formula inspired by the so-called "fractional" expectation threshold [Tal10], again describing the MMSE curve (in this case up to a multiplicative constant) for large enough $n$.

math.ST↗

arXiv2024

Weak Poincar\'e Inequalities, Simulated Annealing, and Sampling from Spherical Spin Glasses

There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincar\'e inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. For example, this occurs if the target distribution has metastable states, small clusters accounting for a vanishing fraction of the mass that are essentially disconnected from the bulk of the measure. Under such conditions, a Poincar\'e inequality cannot hold, necessitating new tools to prove sampling guarantees. We develop a framework to analyze simulated annealing, based on establishing so-called weak Poincar\'e inequalities. These inequalities imply mixing from a suitably warm start, and simulated annealing provides a way to chain such warm starts together into a sampling algorithm. We further identify a local-to-global principle to prove weak Poincar\'e inequalities, mirroring the spectral independence and localization schemes frameworks for analyzing mixing times of Markov chains. As our main application, we prove that simulated annealing samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. This provides the first Markov chain sampling guarantee that holds beyond the uniqueness threshold for spherical spin glasses, where mixing from a worst-case initialization is provably slow due to the presence of metastable states. As an ingredient in our proof, we prove bounds on the operator norm of the covariance matrix of spherical spin glasses in the full replica-symmetric regime. Additionally, we resolve a question related to sampling using data-based initializations.

math.PR↗

arXiv2024

Fast Mixing in Sparse Random Ising Models

Motivated by the community detection problem in Bayesian inference, as well as the recent explosion of interest in spin glasses from statistical physics, we study the classical Glauber dynamics for sampling from Ising models with sparse random interactions. It is now well-known that when the interaction matrix has spectral diameter less than $1$, Glauber dynamics mixes in $O(n\log n)$ steps. Unfortunately, such criteria fail dramatically for interactions supported on arguably the most well-studied sparse random graph: the Erdős--Rényi random graph $G(n,d/n)$, due to the presence of almost linearly many outlier eigenvalues of unbounded magnitude. We prove that for the \emph{Viana--Bray spin glass}, where the interactions are supported on $G(n,d/n)$ and randomly assigned $\pmβ$, Glauber dynamics mixes in $n^{1+o(1)}$ time with high probability as long as $β\le O(1/\sqrt{d})$, independent of $n$. We further extend our results to random graphs drawn according to the $2$-community stochastic block model, as well as when the interactions are given by a "centered" version of the adjacency matrix. The latter setting is particularly relevant for the inference problem in community detection. Indeed, we use this to show that Glauber dynamics succeeds at recovering communities in the stochastic block model in a companion paper [LMR+24]. The primary technical ingredient in our proof is showing that with high probability, a sparse random graph can be decomposed into two parts -- a \emph{bulk} which behaves like a graph with bounded maximum degree and a well-behaved spectrum, and a \emph{near-forest} with favorable pseudorandom properties. We then use this decomposition to design a localization procedure that interpolates to simpler Ising models supported only on the near-forest, and then execute a pathwise analysis to establish a modified log-Sobolev inequality.

math.PR↗

arXiv2024

Locally Stationary Distributions: A Framework for Analyzing Slow-Mixing Markov Chains

Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1. We show that Glauber dynamics on the hardcore model can be used to find independent sets of size $\Omega\left(\frac{\log d}{d} \cdot n\right)$ in triangle-free graphs of degree at most $d$. 2. Let $W$ be a symmetric real matrix with bounded spectral diameter and $v$ be a unit vector. Given the matrix $M = \lambda vv^\top + W$ with a planted rank-one spike along vector $v$, for sufficiently large constant $\lambda$, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the vector $v$. 3. Let $M = A_{\mathbf{G}} - \frac{d}{n}\mathbf{1}\mathbf{1}^\top$ be a centered version of the adjacency matrix where the graph $\mathbf{G}$ is drawn from a sparse 2-community stochastic block model. We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the hidden community vector $\mathbf{\sigma}$.

cs.DS↗

arXiv2022

Sampling from convex sets with a cold start using multiscale decompositions

Running a random walk in a convex body $K\subseteq\mathbb{R}^n$ is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution $\pi_K$ on $K$ after a number of steps polynomial in $n$ and the aspect ratio $R/r$ (i.e., when $rB_2 \subseteq K \subseteq RB_{2}$). Proofs of rapid mixing of such walks often require the probability density $\eta_0$ of the initial distribution with respect to $\pi_K$ to be at most $\mathrm{poly}(n)$: this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein $\eta_0$ can be as high as $\exp(\mathrm{poly}(n))$. Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lov\'asz and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of $\mathbb{R}^n$ into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in $n$ and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for $K$ for a metric that magnifies distances between points close to the boundary of $K$. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of $K$.

cs.DS↗