Hamiltonian Monte Carlo (HMC) is a very popular and generic collection of Markov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity of HMC algorithms is their excellent performance as the dimension $d$ of the target becomes large: under conditions that are satisfied for many common statistical models, optimally-tuned HMC algorithms have a running time that scales like $d^{0.25}$. In stark contrast, the running time of the usual Random-Walk Metropolis (RWM) algorithm, optimally tuned, scales like $d$. This superior scaling of the HMC algorithm with dimension is attributed to the fact that it, unlike RWM, incorporates the gradient information in the proposal distribution. In this paper, we investigate a different scaling question: does HMC beat RWM for highly $\textit{multimodal}$ targets? We find that the answer is often $\textit{no}$. We compute the spectral gaps for both the algorithms for a specific class of multimodal target densities, and show that they are identical. The key reason is that, within one mode, the gradient is effectively ignorant about other modes, thus negating the advantage the HMC algorithm enjoys in unimodal targets. We also give heuristic arguments suggesting that the above observation may hold quite generally. Our main tool for answering this question is a novel simple formula for the conductance of HMC using Liouville's theorem. This result allows us to compute the spectral gap of HMC algorithms, for both the classical HMC with isotropic momentum and the recent Riemannian HMC, for multimodal targets.

Latent position models (LPMs) are a large and popular class of models for random graphs. However, fitting Bayesian LPMs is computationally challenging - computing the likelihood even once takes time that is quadratic in the number of vertices |V| of the observed graph G = (V,E). Many previous papers have introduced approximate MCMC algorithms to speed this up, with the most similar to ours, Rastelli et al (2024), presenting an algorithm that has amortized running time that can be reduced almost to O(|E|) and good empirical performance on reasonable inference problems. The present paper offers two algorithms for solving the same problem: a “fast" algorithm with running time of the same almost-O(|E|) order as astelli et al and much stronger accuracy guarantees, and a “faster" algorithm with an improved running time of almost O(|V|), and accuracy guarantees that are slightly improved compared to Rastelli et al (but not sufficient for all tasks). The main improvements come from the introduction of a simple auxiliary data structure that can be cheaply updated during an MCMC run; we suspect that the same “cheap sketch" may be useful for other MCMC algorithms.

In Diaconis and Saloff-Coste (1996), the authors introduced the simple “transvection" walk on GL_n(𝔽_2): at each step, choose two distinct rows and add one to the other. In Ben-Hamou (2025), the author recently proved that this walk has mixing time O(n^2log n). Inspired by applications in cryptography (see Sotiraki (2016)), Ben-Hamou and Peres (2018) conjectured that the first k columns of this walk mixed in O(nk log(n)) steps. Our main result is a proof of this conjecture uniformly in n and k. Our proof is based on a local-to-global entropy estimate, in the spirit of block factorization results such as Caputo et al (2015), Caputo et al (2021). In our setting, the kernels that correspond roughly to the block kernels of Caputo et al (2021) do not have uniformly large log-Sobolev constants, and so naively applying these techniques does not improve over Ben-Hamou (2025). We avoid these bad blocks by combining our entropy estimates with a burn-in argument similar to classical drift-and-minorization arguments of Rosenthal (1995). This method may be of broader interest, and so we illustrate it by proving an analogous result for a family of product-replacement algorithms on the Heisenberg group.

This chapter surveys progress on three related topics in perturbations of Markov chains: the motivating question of when and how "perturbed" MCMC chains are developed, the theoretical problem of how perturbation theory can be used to analyze such chains, and finally the question of how the theoretical analyses can lead to practical advice.

Motivated by applications in cybersecurity such as finding meaningful sequences of malware-related events buried inside large amounts of computer log data, we introduce the "planted path" problem and propose an algorithm to find fuzzy matchings between two trees. This algorithm can be used as a "building block" for more complicated workflows. We demonstrate usefulness of a few of such workflows in mining synthetically generated data as well as real-world ACME cybersecurity datasets.

Kac's walk on the rotation group, introduced by Hastings in 1970, is an important high-dimensional Markov chain with applications in statistical physics, statistics, cryptography, and computational science. Despite its simple transition rules, determining its total-variation mixing time has remained a challenging problem for decades. A key obstacle is that the walk is not conjugation-invariant, placing it beyond the reach of classical Fourier-analytic techniques that apply to many related random walks on compact groups. We prove that Kac's walk mixes in total variation in O(n^2 log n) steps, matching the conjectured mixing time up to constants. The proof is based on a refined two-stage coupling. Building on earlier work, the first stage contracts two copies of the chain to a small neighborhood via a Wasserstein coupling. Our main contribution is a new framework for analyzing the second-stage coupling. It can be viewed as a discrete analogue of Malliavin calculus for Markov chains. We represent the law of the chain as the pushforward of high-dimensional noise and prove quantitative non-degeneracy of the associated linearization using matrix martingale methods. This yields an approximately Gaussian distribution in the Lie algebra with well-conditioned covariance, allowing small group translations to be absorbed at negligible cost in total variation. Our approach provides a general framework for studying mixing in high-dimensional Markov chains in continuous state spaces with singular transition kernels.

In latent-position random graph models (LPMs), latent vertex positions U_1,…,U_n are sampled from some distribution on a latent space Ω, then edges of an observed graph G = ([n],E) are sampled with some probability ℙ[(i,j) ∈ E ]=w(U_i,U_j) that depends on the unobserved latent positions. LPMs are ubiquitous in the statistical analysis of networks, offering models that have good empirical performance, strong theoretical guarantees, and tractable algorithms. The special case Ω= [0,1] is important, as it corresponds to graphs with temporal or preference-based structure. In this paper, we study three problems related to LPMs with latent space [0,1]: ordering the vertices according to the latent positions, estimating the generating graphon w, and testing whether an observed graph G could have come from an LPM with state space [0,1]. Our results on the ordering problem greatly generalize two observations of Janssen/Smith (2022): (i) for some families of graphons, the best estimate of the ordering converges much faster than the usual statistical rate of 1/√(n), and (ii) this occurs even though, for the same families of graphons, the best estimate of the latent positions still occurs at the usual 1/√(n) rate. As a main consequence, we develop a computationally-efficient graphon-estimation algorithm and show that it has the same convergence rate as the non-explicit optimal algorithm of Gao et al (2015). We also derive and analyze a testing procedure.

Bayesian multidimensional scaling (BMDS) embeds n objects in a low-dimensional space to approximately preserve an observed dissimilarity matrix. Compared to classic MDS, BMDS is more robust to model misspecification and supports posterior uncertainty quantification and joint estimation within hierarchical models. However, standard BMDS inference is computationally prohibitive, requiring O(n^2) operations per MCMC iteration to evaluate the likelihood. We propose Barnes–Hut BMDS (BH-BMDS), which uses a tree-based approximation to the likelihood and a Gibbs sampler that leverages this structure, remaining compatible with hierarchical extensions. BH-BMDS reduces computational complexity to O(n log n) while preserving the geometric fidelity of the embedding. We further establish consistency for the stationary measure of BH-BMDS, proving that it concentrates around the true latent configuration even as the total error of the surrogate likelihood diverges. Notably, this consistency holds in the infinite-dimensional limit. We evaluate the approximation on datasets with diverse structure, including air traffic networks, arXiv abstracts, MNIST images and neural activity recordings from mouse models of tau pathology. Across all settings, BH-BMDS closely matches BMDS while achieving substantial computational gains, with approximately 10-fold speedups at n=1,000 and 70-fold speedups at n=10,000. These gains increase with n, demonstrating strong empirical scalability.

Consider a random graph $G$ of size $N$ constructed according to a \textit{graphon} $w \, : \, [0,1]^{2} \mapsto [0,1]$ as follows. First embed $N$ vertices $V = \{v_1, v_2, \ldots, v_N\}$ into the interval $[0,1]$, then for each $i < j$ add an edge between $v_{i}, v_{j}$ with probability $w(v_{i}, v_{j})$. Given only the adjacency matrix of the graph, we might expect to be able to approximately reconstruct the permutation $\sigma$ for which $v_{\sigma(1)} < \ldots < v_{\sigma(N)}$ if $w$ satisfies the following \textit{linear embedding} property introduced in [Janssen 2019]: for each $x$, $w(x,y)$ decreases as $y$ moves away from $x$. For a large and non-parametric family of graphons, we show that (i) the popular spectral seriation algorithm [Atkins 1998] provides a consistent estimator $\hat{\sigma}$ of $\sigma$, and (ii) a small amount of post-processing results in an estimate $\tilde{\sigma}$ that converges to $\sigma$ at a nearly-optimal rate, both as $N \rightarrow \infty$.

Motivated by a conjecture of Vaikuntanathan and Zamir, we study the pseudo-mixing of Kac's walk on SO(n): whether short trajectories are indistinguishable from Haar measure by low-complexity tests. We prove that the first k columns mix in Wasserstein distance in O(n(k+log n)log n) steps for fixed accuracy, resolving a conjecture of Oliveira. Combining this with a representation-theoretic variance bound, we show that if T=ω(nk(k+log n)log n), then every degree-k polynomial normalized to have unit Haar variance has expectation under the T-step law within o(1) of its Haar expectation. As an application, we show that this pseudo-mixing estimate can be used to prove the effectiveness of a fast Johnson–Lindenstrauss transform with the usual target dimension.

As sample sizes grow, scalability has become a central concern in the development of Markov chain Monte Carlo (MCMC) methods. One general approach to this problem, exemplified by the popular stochastic gradient Langevin dynamics (SGLD) algorithm, is to use a small random subsample of the data at every time step. This paper shows that this approach often fails: while decreasing the sample size increases the speed of each MCMC step, for typical datasets this is balanced by a matching decrease in accuracy. This result complements recent work on lower bounds for subsampling MCMC by focusing on actual errors for SGLD, including non-reversible algorithms, and by giving assumptions that are comparatively easy to verify.

We show the asymptotic non-equivalence of a simple diffusion and its Euler scheme experiment when the variance is not known. This negative result contrasts sharply with the case of a known diffusion coefficient, where Genon-Catalot and Laredo (2014) established asymptotic equivalence between discrete and continuous experiments. Informally, this means that observing only the Euler discretization entails losing information compared to the fully-observed process in this case. In supplementary material, we extend the comparison to some nonparametric classes of diffusions.

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion‐like processes. In this paper, we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well‐approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we compare the Dirichlet forms of two general Markov processes by applying the transfer of the well‐known comparison theorem for finite Markov processes.

Bayesian multidimensional scaling (BMDS) is a probabilistic dimension reduction tool that allows one to model and visualize data consisting of dissimilarities between pairs of objects. Although BMDS has proven useful within, e.g., Bayesian phylogenetic inference, its likelihood and gradient calculations require a burdensome order of N^2 floating-point operations, where N is the number of data points. Thus, BMDS becomes impractical as N grows large. We propose and compare two sparse versions of BMDS (sBMDS) that apply log-likelihood and gradient computations to subsets of the observed dissimilarity matrix data. Landmark sBMDS (L-sBMDS) extracts columns, while banded sBMDS (B-sBMDS) extracts diagonals of the data. These sparse variants let one specify a time complexity between N^2 and N. Under simplified settings, we prove posterior consistency for subsampled distance matrices. Through simulations, we examine the accuracy and computational efficiency across all models using both the Metropolis-Hastings and Hamiltonian Monte Carlo algorithms. We observe approximately 3-fold, 10-fold and 40-fold speedups with negligible loss of accuracy, when applying the sBMDS likelihoods and gradients to 500, 1,000 and 5,000 data points with 50 bands (landmarks); these speedups only increase with the size of data considered. Finally, we apply the sBMDS variants to the phylogeographic modeling of multiple influenza subtypes to better understand how these strains spread through global air transportation networks.

The basic question in perturbation analysis of Markov chains is: how do small changes in the transition kernels of Markov chains translate to chains in their stationary distributions? Many papers on the subject have shown, roughly, that the change in stationary distribution is small as long as the change in the kernel is much less than some measure of the convergence rate. This result is essentially sharp for generic Markov chains. In this paper we show that much larger errors, up to size roughly the square root of the convergence rate, are permissible for many target distributions associated with graphical models. The main motivation for this work comes from computational statistics, where there is often a tradeoff between the per-step error and per-step cost of approximate MCMC algorithms. Our results show that larger perturbations (and thus less-expensive chains) still give results with small error.

Community detection methods play a central role in understanding complex networks by revealing highly connected subsets of entities. However, most community detection algorithms generate partitions of the nodes, thus (i) forcing every node to be part of a community and (ii) ignoring the possibility that some nodes may be part of multiple communities. In our work, we investigate three simple community association strength (CAS) scores and their usefulness as post-processing tools given some partition of the nodes. We show that these measures can be used to improve node partitions, detect outlier nodes (not part of any community), and help find nodes with multiple community memberships.

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster, more interpretable, and can be investigated analytically. In this paper, we use the underlying ingredients of the ABCD model, and its generalization to include outliers (ABCD+o), and introduce another variant that allows for overlapping communities, ABCD+o^2.

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but is faster and more interpretable. In this paper, we use the underlying ingredients of the ABCD model, and its generalization to include outliers (ABCD+o), and introduce another variant for overlapping communities, 𝐀𝐁𝐂𝐃+𝐨^2 .

We investigate the sharpness of the spectral profile bound presented by Goel et al. and Chen et al. on the L^2 mixing time of Markov chains on continuous state spaces. We show that the bound provided by Chen et al. is sharp up to a factor of loglog of the initial density. This result extends the findings of Kozma, which showed the analogous result for the original spectral profile bound of Goel et al. for Markov chains on finite state spaces. Kozma shows that the spectral profile bound is sharp up to a multiplicative factor of log(log(π_min)), where π_min is the smallest value of the probability mass function of the stationary distribution. We discuss the application of our primary finding to the comparison of Markov chains. Our main result can be used as a comparison bound, indicating that it is possible to compare chains even when only non-spectral bounds exist for a known chain.

BackgroundElectronic health records are a valuable source of patient information that must be properly deidentified before being shared with researchers. This process requires expertise and time. In addition, synthetic data have considerably reduced the restrictions on the use and sharing of real data, allowing researchers to access it more rapidly with far fewer privacy constraints. Therefore, there has been a growing interest in establishing a method to generate synthetic data that protects patients’ privacy while properly reflecting the data. ObjectiveThis study aims to develop and validate a model that generates valuable synthetic longitudinal health data while protecting the privacy of the patients whose data are collected. MethodsWe investigated the best model for generating synthetic health data, with a focus on longitudinal observations. We developed a generative model that relies on the generalized canonical polyadic (GCP) tensor decomposition. This model also involves sampling from a latent factor matrix of GCP decomposition, which contains patient factors, using sequential decision trees, copula, and Hamiltonian Monte Carlo methods. We applied the proposed model to samples from the MIMIC-III (version 1.4) data set. Numerous analyses and experiments were conducted with different data structures and scenarios. We assessed the similarity between our synthetic data and the real data by conducting utility assessments. These assessments evaluate the structure and general patterns present in the data, such as dependency structure, descriptive statistics, and marginal distributions. Regarding privacy disclosure, our model preserves privacy by preventing the direct sharing of patient information and eliminating the one-to-one link between the observed and model tensor records. This was achieved by simulating and modeling a latent factor matrix of GCP decomposition associated with patients. ResultsThe findings show that our model is a promising method for generating synthetic longitudinal health data that is similar enough to real data. It can preserve the utility and privacy of the original data while also handling various data structures and scenarios. In certain experiments, all simulation methods used in the model produced the same high level of performance. Our model is also capable of addressing the challenge of sampling patients from electronic health records. This means that we can simulate a variety of patients in the synthetic data set, which may differ in number from the patients in the original data. ConclusionsWe have presented a generative model for producing synthetic longitudinal health data. The model is formulated by applying the GCP tensor decomposition. We have provided 3 approaches for the synthesis and simulation of a latent factor matrix following the process of factorization. In brief, we have reduced the challenge of synthesizing massive longitudinal health data to synthesizing a nonlongitudinal and significantly smaller data set.