arXiv subjects

Xusheng Zhang

Publications and source records attributed to Xusheng Zhang.

12 recordsLinked to original sources

arXiv2026

Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma

The Lovász Local Lemma (LLL) is a probabilistic tool that has been shown to be of central importance in the study of distributed algorithms. For example, the constructive LLL is known to be complete for the class of locally-checkable labeling problems with $o(\log n)$ randomized complexities in the LOCAL model. One classic application of the LLL is in coloring graphs with some sparse structure, such as triangle-free graphs. Triangle-free coloring therefore serves as a benchmark problem for techniques for sublogarithmic randomized distributed algorithms. The state-of-the-art distributed triangle-free coloring algorithm of Pettie and Su [ICALP 2013, Information and Computation 2015] uses $\fracΔ{k}$ colors (where $k$ can be up to $(\frac14 - \varepsilon)\ln Δ$) and consists of $O(k+\log^* n)$ applications of the distributed LLL. However, the distributed LLL is itself a difficult problem; despite significant study, the fastest algorithms known require $O(\log_Δn)$ or $O(\fracΔ{\logΔ})+\log^{O(1)}\log n$ rounds. In this work, we adapt the Pettie-Su's algorithm so that the resulting LLL instances can be solved in $\log^{O(1)}\log n$ rounds, by employing the 'resilience' definition of Davies [SODA 2023]. This gives an $O(k)+ \log^{O(1)}\log n$ complexity (since the LLL is not needed when $k= \log^{ω(1)}\log n$), essentially causing the LLL steps to no longer be the bottleneck of the algorithm. As a corollary we obtain the first $\log^{O(1)}\log n$-round algorithms for coloring triangle-free graphs with $o(Δ)$ colors. The same framework also yields a companion girth-$5$ algorithm, using $(1+\varepsilon)Δ/\ln Δ$ colors in $O(k)+ \log^{O(1)}\log n$ rounds, matching the best known existential upper bound for the number of colors.

cs.DC↗

arXiv2026

Teaching Foundation Models to Read mmWave: Pose-Guided Kinematic Representation for Human Behavior Understanding

Large language model agents need to perceive human behavior in physical environments. Millimeter-wave (mmWave) radar provides a privacy-friendly and contactless sensing modality, but radar observations are difficult to align with language. Existing radar-language methods often rely on synthetic data or lack explicit supervision for human body structure and motion. We present mmMind, a radar-language model that uses synchronized 3D pose as training-only supervision. A spatio-temporal radar encoder is pretrained to capture body configuration and motion dynamics, after which the pose head is removed so that inference requires radar alone. The learned radar representations are then aligned with an LLM for behavior captioning and spatio-temporal question answering. We also introduce mmMind-Bench, a real-world mmWave-language benchmark containing 17.9 hours of recordings from 23 participants across seven indoor environments. Experiments on captioning, question answering, and unseen-action generalization show that mmMind consistently outperforms existing radar-language baselines, while ablations confirm the importance of pose-guided pretraining.

cs.CV↗

arXiv2026

Fast Mixing for Low-Temperature Potts Models via Poisson Trees

The $q$-state ferromagnetic Potts model on a graph $G$ is a probability distribution on all $q$-colourings of $G$ that favours many monochromatic edges. Approximate sampling from the Potts model is a central problem in the study of spin systems on sparse graphs, especially in the low-temperature regime, where the model strongly favours ordered configurations, often creating bottlenecks that make Markov-chain sampling inefficient or difficult to analyse. We focus on the sparse random graph $G(n,d/n)$. The local neighbourhoods of $G(n,d/n)$ are tree-like, but the relevant underlying graph is a Poisson Galton-Watson tree. This motivates the study of Glauber dynamics for the low-temperature Potts model on such trees with monochromatic boundary conditions. The Poisson setting introduces difficulties absent from the regular case: degrees fluctuate, long induced paths may appear, and branches can terminate before reaching the boundary. As a result, the effect of the monochromatic boundary at the leaves is much less uniform. Our main result shows near-linear mixing for the Glauber dynamics on Poisson trees with monochromatic boundary conditions. This extends the corresponding regular-tree results of Martinelli, Sinclair, and Weitz (SODA 2004) and of Blanca, Chen, Stefankovič, and Vigoda (RANDOM 2021) to the irregular trees arising from sparse random graphs. Our proof introduces an adaptive block decomposition of the tree, built around regions containing large regular subtrees, and combines it with correlation-decay estimates and functional-inequality arguments. We also obtain a near-linear-time approximate sampling algorithm for the Potts model on $G(n,d/n)$ at all temperatures, speeding up the best previous algorithm of Galanis, Goldberg, and Smolarova (ICALP 2025). The main new ingredient is a refined analysis of the low-temperature regime, building on the Poisson tree result.

math.PR↗

arXiv2026

Time-varying Mixing Matrix Design for Energy-efficient Decentralized Federated Learning

We consider the design of mixing matrices to minimize the operation cost for decentralized federated learning (DFL) in wireless networks, with focus on minimizing the maximum per-node energy consumption. As a critical hyperparameter for DFL, the mixing matrix controls both the convergence rate and the needs of agent-to-agent communications, and has thus been studied extensively. However, existing designs mostly focused on minimizing the communication time, leaving open the minimization of per-node energy consumption that is critical for energy-constrained devices. This work addresses this gap through a theoretically-justified solution for mixing matrix design that aims at minimizing the maximum per-node energy consumption until convergence, while taking into account the broadcast nature of wireless communications. Based on a novel convergence theorem that allows arbitrarily time-varying mixing matrices, we propose a multi-phase design framework that activates time-varying communication topologies under optimized budgets to trade off the per-iteration energy consumption and the convergence rate while balancing the energy consumption across nodes. Our evaluations based on real data have validated the efficacy of the proposed solution in combining the low energy consumption of sparse mixing matrices and the fast convergence of dense mixing matrices.

cs.LG↗

arXiv2026

Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs

We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, approximately sampling from this model on graphs of degree at most $Δ$ is computationally hard. At parameter $p$ below the tree uniqueness threshold $p_{\mathsf{u}}(q,Δ)$, it is expected that sampling is easy and local Markov chains mix rapidly on all bounded degree graphs. On typical graphs (e.g., random regular graphs), the same is predicted at $p > p_{\mathsf{s}}(q,Δ)$, where $p_{\mathsf{s}}(q,Δ)$ is a second uniqueness transition point on the $Δ$-regular wired tree. Our first result establishes this non-uniqueness/uniqueness phase transition at $p_{\mathsf{s}}(q,Δ)$ for all $q$ on the infinite $Δ$-regular wired tree, resolving a conjecture of H{ä}ggstr{ö}m (1996). For this, we establish weak spatial mixing at $p>p_{\mathsf{s}}(q,Δ)$ under sufficiently wired boundary conditions. We use this understanding of decay of correlations to show that on the wired tree on $n$ vertices, whenever $q>1$ and $p>p_{\mathsf{s}}(q,Δ)$, the mixing time of random-cluster Glauber dynamics is a near-optimal $n^{1+o(1)}$. We then extend these results on spatial and temporal mixing from the tree to treelike geometries with mostly wired boundaries and use them to show that the random-cluster Glauber dynamics mix rapidly on the random $Δ$-regular graph for all $p>p_{\mathsf{s}}(q,Δ)$ as long as $q \ge C \log Δ$, providing an efficient sampling algorithm for both the random-cluster and Potts models in this context.

math.PR↗

arXiv2025

One-Shot Learning for k-SAT

Consider a $k$-SAT formula $Φ$ where every variable appears at most $d$ times. Let $σ$ be a satisfying assignment, sampled proportionally to $e^{βm(σ)}$ where $m(σ)$ is the number of true variables and $β$ is a real parameter. Given $Φ$ and $σ$, can we efficiently learn $β$? This problem falls into a recent line of work about single-sample (``one-shot'') learning of Markov random fields. Our $k$-SAT setting was recently studied by Galanis, Kalavasis, Kandiros (SODA24). They showed that single-sample learning is possible when roughly $d\leq 2^{k/6.45}$ and impossible when $d\geq (k+1) 2^{k-1}$. In addition to the gap in~$d$, their impossibility result left open the question of whether the feasibility threshold for one-shot learning is dictated by the satisfiability threshold for bounded-degree $k$-SAT formulas. Our main contribution is to answer this question negatively. We show that one-shot learning for $k$-SAT is infeasible well below the satisfiability threshold; in fact, we obtain impossibility results for degrees $d$ as low as $k^2$ when $β$ is sufficiently large, and bootstrap this to small values of $β$ when $d$ scales exponentially with $k$, via a probabilistic construction. On the positive side, we simplify the analysis of the learning algorithm, obtaining significantly stronger bounds on $d$ in terms of $β$. For the uniform case $β\rightarrow 0$, we show that learning is possible under the condition $d\lesssim 2^{k/2}$. This is (up to constant factors) all the way to the sampling threshold -- it is known that sampling a uniformly-distributed satisfying assignment is NP-hard for $d\gtrsim 2^{k/2}$.

cs.DS↗

arXiv2025

Mean-field Potts and random-cluster dynamics from high-entropy initializations

A common obstruction to efficient sampling from high-dimensional distributions with Markov chains is the multimodality of the target distribution because they may get trapped far from stationarity. Still, one hopes that this is only a barrier to the mixing of Markov chains from worst-case initializations and can be overcome by choosing high-entropy initializations, e.g., a product or weakly correlated distribution. Ideally, from such initializations, the dynamics would escape from the saddle points separating modes quickly and spread its mass between the dominant modes with the correct probabilities. In this paper, we study convergence from high-entropy initializations for the random-cluster and Potts models on the complete graph -- two extensively studied high-dimensional landscapes that pose many complexities like discontinuous phase transitions and asymmetric metastable modes. We study the Chayes--Machta and Swendsen--Wang dynamics for the mean-field random-cluster model and the Glauber dynamics for the Potts model. We sharply characterize the set of product measure initializations from which these Markov chains mix rapidly, even though their mixing times from worst-case initializations are exponentially slow. Our proofs require careful approximations of projections of high-dimensional Markov chains (which are not themselves Markovian) by tractable 1-dimensional random processes, followed by analysis of the latter's escape from saddle points separating stable modes.

math.PR↗

arXiv2024

Energy-efficient Decentralized Learning via Graph Sparsification

This work aims at improving the energy efficiency of decentralized learning by optimizing the mixing matrix, which controls the communication demands during the learning process. Through rigorous analysis based on a state-of-the-art decentralized learning algorithm, the problem is formulated as a bi-level optimization, with the lower level solved by graph sparsification. A solution with guaranteed performance is proposed for the special case of fully-connected base topology and a greedy heuristic is proposed for the general case. Simulations based on real topology and dataset show that the proposed solution can lower the energy consumption at the busiest node by 54%-76% while maintaining the quality of the trained model.

cs.LG↗

arXiv2023

Diameters of symmetric and lifted simple exclusion models

We determine diameters of Markov chains describing one-dimensional N -particle models with an exclusion interaction, namely the Ssep (symmetric simple exclusion process) and one of its non-reversible liftings, the lifted Tasep (totally asymmetric simple exclusion process). The diameters provide lower bounds for the mixing times, and we discuss the implications of our findings for the analysis of these models.

cond-mat.stat-mech↗

arXiv2023

Rapid mixing of global Markov chains via spectral independence: the unbounded degree case

We consider spin systems on general $n$-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen--Wang dynamics for the $q$-state ferromagnetic Potts model on graphs of maximum degree $Δ$, where $Δ$ is allowed to grow with $n$, converges in $O((Δ\log n)^c)$ steps where $c > 0$ is a constant independent of $Δ$ and $n$. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is $O(Δ^c \log n)$ for a constant $c>0$ independent of $Δ$ and $n$. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. Our result implies optimal $O(\log n)$ mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called $k$-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

math.PR↗

arXiv2022

The Critical Mean-field Chayes-Machta Dynamics

The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on $n$ vertices. The random-cluster model is parametrized by an edge probability $p$ and a cluster weight $q$. Our focus is on the critical regime: $p = p_c(q)$ and $q \in (1,2)$, where $p_c(q)$ is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is $O(\log n \cdot \log \log n)$ in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the $\log\log n$ factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

math.PR↗

arXiv2008

Stability analysis of twist grain boundaries in lamellar phases of block copolymers

Twist grain boundaries are widely observed in lamellar phases of block copolymers. A mesoscopic model of the copolymer is used to obtain stationary configurations that include a twist grain boundary, and to analyze their stability against long wavelength perturbations. The analysis presented is valid in the weak segregation regime, and includes direct numerical solution of the governing equations as well as a multiple scale analysis. We find that a twist boundary configuration with arbitrary misorientation angle can be well described by two modes, and obtain the equations for their slowly varying amplitudes. The width of the boundary region is seen to scale as $ε^{-1/4}$, with $ε$ being the dimensionless distance to the order-disorder transition. We finally present the results of the linear stability analysis of the planar boundary.

cond-mat.soft↗