arXiv subjects

Leo L. Duan

Publications and source records attributed to Leo L. Duan.

At least 19 recordsLinked to original sources

arXiv2026

Consistency of Graphical Model-based Clustering: Robust Clustering using Bayesian Spanning Forest

Mixture model-based frameworks are very popular for statistical inference in clustering. While convenient for producing probabilistic estimates of cluster assignments and uncertainty, they are prone to misspecification, which can lead to inconsistent clustering results. Graphical model-based clustering adopts a different strategy, specifying the likelihood by treating data as dependently generated from a disjoint union of component graphs. Recent work on Bayesian spanning forests addresses graph uncertainty by using the integrated posterior of the node partition, marginalized over the latent edge distribution, to produce probabilistic clustering estimates. Despite strong empirical performance, theoretical guarantees such as consistency remain unclear, particularly when the true data-generating process deviates from the assumed graphical model. This article establishes a positive asymptotic result: when data are generated from an unknown collection of component distributions and a mild asymptotic separation condition holds with probability tending to one (without requiring complete support separation), the posterior concentrates on the true partition, thereby yielding consistent clustering estimates, including the number of clusters. Our results hold whether the number of clusters is fixed or increases with sample size. Additionally, we derive an upper bound on the expected misclassification rate. These results highlight graphical models as a robust alternative to mixture models in clustering.

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arXiv2025

Graphical Model-based Inference on Persistent Homology

Persistent homology is a cornerstone of topological data analysis, offering a multiscale summary of topology with robustness to nuisance transformations, such as rotations and small deformations. Persistent homology has seen broad use across domains such as computer vision and neuroscience. Most statistical treatments, however, use homology primarily as a feature extractor, relying on statistical distance-based tests or simple time-to-event models for inferential tasks. While these approaches can detect global differences, they rarely localize the source of those differences. We address this gap by taking a graphical model-based approach: we associate each vertex with a population latent position in a conic space and model each bar's key events (birth and death times) using an exponential distribution, whose rate is a transformation of the latent positions according to an event occurring on the graph. The low-dimensional bars have simple graph-event representations, such as the formation of a minimum spanning tree or the triangulation of a loop, and thus enjoy tractable likelihoods. Taking a Bayesian approach, we infer latent positions and enable model extensions such as hierarchical models that allow borrowing strength across groups. Applications to a neuroimaging study of Alzheimer's disease demonstrate that our method localizes sources of difference and provides interpretable, model-based analyses of topological structure in complex data. The code is provided and maintained at https://github.com/zitian-wu/graphPH.

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arXiv2025

Exact Sampling of Spanning Trees via Fast-forwarded Random Walks

Tree graphs are routinely used in statistics. When estimating a Bayesian model with a tree component, sampling the posterior remains a core difficulty. Existing Markov chain Monte Carlo methods tend to rely on local moves, often leading to poor mixing. A promising approach is to instead directly sample spanning trees on an auxiliary graph. Current spanning tree samplers, such as the celebrated Aldous--Broder algorithm, predominantly rely on simulating random walks that are required to visit all the nodes of the graph. Such algorithms are prone to getting stuck in certain sub-graphs. We formalize this phenomenon using the bottlenecks in the random walk's transition probability matrix. We then propose a novel fast-forwarded cover algorithm that can break free from bottlenecks. The core idea is a marginalization argument that leads to a closed-form expression which allows for fast-forwarding to the event of visiting a new node. Unlike many existing approximation algorithms, our algorithm yields exact samples. We demonstrate the enhanced efficiency of the fast-forwarded cover algorithm, and illustrate its application in fitting a Bayesian dendrogram model on a Massachusetts crimes and communities dataset.

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arXiv2024

Graph-accelerated Markov Chain Monte Carlo using Approximate Samples

It has become increasingly easy nowadays to collect approximate posterior samples via fast algorithms such as variational Bayes, but concerns exist about the estimation accuracy. It is tempting to build solutions that exploit approximate samples in a canonical Markov chain Monte Carlo framework. A major barrier is that the approximate sample as a proposal tends to have a low Metropolis-Hastings acceptance rate, as the dimension increases. In this article, we propose a simple solution named graph-accelerated Markov Chain Monte Carlo. We build a graph with each node assigned to an approximate sample, then run Markov chain Monte Carlo with random walks over the graph. We optimize the graph edges to enforce small differences in posterior density/probability between nodes, while encouraging edges to have large distances in the parameter space. The graph allows us to accelerate a canonical Markov transition kernel through mixing with a large-jump Metropolis-Hastings step. The acceleration is easily applicable to existing Markov chain Monte Carlo algorithms. We theoretically quantify the rate of acceptance as dimension increases, and show the effects on improved mixing time. We demonstrate improved mixing performances for challenging problems, such as those involving multiple modes, non-convex density contour, or large-dimension latent variables.

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arXiv2024

Gibbs Sampling using Anti-correlation Gaussian Data Augmentation, with Applications to L1-ball-type Models

L1-ball-type priors are a recent generalization of the spike-and-slab priors. By transforming a continuous precursor distribution to the L1-ball boundary, it induces exact zeros with positive prior and posterior probabilities. With great flexibility in choosing the precursor and threshold distributions, we can easily specify models under structured sparsity, such as those with dependent probability for zeros and smoothness among the non-zeros. Motivated to significantly accelerate the posterior computation, we propose a new data augmentation that leads to a fast block Gibbs sampling algorithm. The latent variable, named ``anti-correlation Gaussian'', cancels out the quadratic exponent term in the latent Gaussian distribution, making the parameters of interest conditionally independent so that they can be updated in a block. Compared to existing algorithms such as the No-U-Turn sampler, the new blocked Gibbs sampler has a very low computing cost per iteration and shows rapid mixing of Markov chains. We establish the geometric ergodicity guarantee of the algorithm in linear models. Further, we show useful extensions of our algorithm for posterior estimation of general latent Gaussian models, such as those involving multivariate truncated Gaussian or latent Gaussian process.

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arXiv2023

Low Tree-Rank Bayesian Vector Autoregression Model

Vector autoregression has been widely used for modeling and analysis of multivariate time series data. In high-dimensional settings, model parameter regularization schemes inducing sparsity yield interpretable models and achieved good forecasting performance. However, in many data applications, such as those in neuroscience, the Granger causality graph estimates from existing vector autoregression methods tend to be quite dense and difficult to interpret, unless one compromises on the goodness-of-fit. To address this issue, this paper proposes to incorporate a commonly used structural assumption -- that the ground-truth graph should be largely connected, in the sense that it should only contain at most a few components. We take a Bayesian approach and develop a novel tree-rank prior distribution for the regression coefficients. Specifically, this prior distribution forces the non-zero coefficients to appear only on the union of a few spanning trees. Since each spanning tree connects $p$ nodes with only $(p-1)$ edges, it effectively achieves both high connectivity and high sparsity. We develop a computationally efficient Gibbs sampler that is scalable to large sample size and high dimension. In analyzing test-retest functional magnetic resonance imaging data, our model produces a much more interpretable graph estimate, compared to popular existing approaches. In addition, we show appealing properties of this new method, such as efficient computation, mild stability conditions and posterior consistency.

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arXiv2023

Spectral Clustering, Bayesian Spanning Forest, and Forest Process

Spectral clustering views the similarity matrix as a weighted graph, and partitions the data by minimizing a graph-cut loss. Since it minimizes the across-cluster similarity, there is no need to model the distribution within each cluster. As a result, one reduces the chance of model misspecification, which is often a risk in mixture model-based clustering. Nevertheless, compared to the latter, spectral clustering has no direct ways of quantifying the clustering uncertainty (such as the assignment probability), or allowing easy model extensions for complicated data applications. To fill this gap, we propose the Bayesian forest model as a generative graphical model for spectral clustering. This is motivated by our discovery that the posterior connecting matrix in a forest model has almost the same leading eigenvectors, as the ones used by normalized spectral clustering. To induce a distribution for the forest, we develop a ``forest process'' as a graph extension to the urn process, while we carefully characterize the differences in the partition probability. We derive a simple Markov chain Monte Carlo algorithm for posterior estimation, and demonstrate superior performance compared to existing algorithms. We illustrate several model-based extensions useful for data applications, including high-dimensional and multi-view clustering for images.

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arXiv2023

Bayesian Inference with the l1-ball Prior: Solving Combinatorial Problems with Exact Zeros

The l1-regularization is very popular in high dimensional statistics -- it changes a combinatorial problem of choosing which subset of the parameter are zero, into a simple continuous optimization. Using a continuous prior concentrated near zero, the Bayesian counterparts are successful in quantifying the uncertainty in the variable selection problems; nevertheless, the lack of exact zeros makes it difficult for broader problems such as the change-point detection and rank selection. Inspired by the duality of the l1-regularization as a constraint onto an l1-ball, we propose a new prior by projecting a continuous distribution onto the l1-ball. This creates a positive probability on the ball boundary, which contains both continuous elements and exact zeros. Unlike the spike-and-slab prior, this l1-ball projection is continuous and differentiable almost surely, making the posterior estimation amenable to the Hamiltonian Monte Carlo algorithm. We examine the properties, such as the volume change due to the projection, the connection to the combinatorial prior, the minimax concentration rate in the linear problem. We demonstrate the usefulness of exact zeros that simplify the combinatorial problems, such as the change-point detection in time series, the dimension selection of mixture model and the low-rank-plus-sparse change detection in the medical images.

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arXiv2023

Consistent Model-based Clustering: using the Quasi-Bernoulli Stick-Breaking Process

In mixture modeling and clustering applications, the number of components and clusters is often not known. A stick-breaking mixture model, such as the Dirichlet process mixture model, is an appealing construction that assumes infinitely many components, while shrinking the weights of most of the unused components to near zero. However, it is well-known that this shrinkage is inadequate: even when the component distribution is correctly specified, spurious weights appear and give an inconsistent estimate of the number of clusters. In this article, we propose a simple solution: when breaking each mixture weight stick into two pieces, the length of the second piece is multiplied by a quasi-Bernoulli random variable, taking value one or a small constant close to zero. This effectively creates a soft-truncation and further shrinks the unused weights. Asymptotically, we show that as long as this small constant diminishes to zero at a rate faster than $o(1/n^2)$, with $n$ the sample size, the posterior distribution will converge to the true number of clusters. In comparison, we rigorously explore Dirichlet process mixture models using a concentration parameter that is either constant or rapidly diminishes to zero -- both of which lead to inconsistency for the number of clusters. Our proposed model is easy to implement, requiring only a small modification of a standard Gibbs sampler for mixture models. In simulations and a data application of clustering brain networks, our proposed method recovers the ground-truth number of clusters, and leads to a small number of clusters.

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arXiv2022

Bayesian Inference using the Proximal Mapping: Uncertainty Quantification under Varying Dimensionality

In statistical applications, it is common to encounter parameters supported on a varying or unknown dimensional space. Examples include the fused lasso regression, the matrix recovery under an unknown low rank, etc. Despite the ease of obtaining a point estimate via the optimization, it is much more challenging to quantify their uncertainty -- in the Bayesian framework, a major difficulty is that if assigning the prior associated with a $p$-dimensional measure, then there is zero posterior probability on any lower-dimensional subset with dimension $d<p$; to avoid this caveat, one needs to choose another dimension-selection prior on $d$, which often involves a highly combinatorial problem. To significantly reduce the modeling burden, we propose a new generative process for the prior: starting from a continuous random variable such as multivariate Gaussian, we transform it into a varying-dimensional space using the proximal mapping. This leads to a large class of new Bayesian models that can directly exploit the popular frequentist regularizations and their algorithms, such as the nuclear norm penalty and the alternating direction method of multipliers, while providing a principled and probabilistic uncertainty estimation. We show that this framework is well justified in the geometric measure theory, and enjoys a convenient posterior computation via the standard Hamiltonian Monte Carlo. We demonstrate its use in the analysis of the dynamic flow network data.

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arXiv2021

Bayesian Spanning Tree: Estimating the Backbone of the Dependence Graph

In multivariate data analysis, it is often important to estimate a graph characterizing dependence among (p) variables. A popular strategy uses the non-zero entries in a (p\times p) covariance or precision matrix, typically requiring restrictive modeling assumptions for accurate graph recovery. To improve model robustness, we instead focus on estimating the {\em backbone} of the dependence graph. We use a spanning tree likelihood, based on a minimalist graphical model that is purposely overly-simplified. Taking a Bayesian approach, we place a prior on the space of trees and quantify uncertainty in the graphical model. In both theory and experiments, we show that this model does not require the population graph to be a spanning tree or the covariance to satisfy assumptions beyond positive-definiteness. The model accurately recovers the backbone of the population graph at a rate competitive with existing approaches but with better robustness. We show combinatorial properties of the spanning tree, which may be of independent interest, and develop an efficient Gibbs sampler for Bayesian inference. Analyzing electroencephalography data using a Hidden Markov Model with each latent state modeled by a spanning tree, we show that results are much more interpretable compared with popular alternatives.

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arXiv2021

Transport Monte Carlo: High-Accuracy Posterior Approximation via Random Transport

In Bayesian applications, there is a huge interest in rapid and accurate estimation of the posterior distribution, particularly for high dimensional or hierarchical models. In this article, we propose to use optimization to solve for a joint distribution (random transport plan) between two random variables, $θ$ from the posterior distribution and $β$ from the simple multivariate uniform. Specifically, we obtain an approximate estimate of the conditional distribution $Π(β\mid θ)$ as an infinite mixture of simple location-scale changes; applying the Bayes' theorem, $Π(θ\midβ)$ can be sampled as one of the reversed transforms from the uniform, with the weight proportional to the posterior density/mass function. This produces independent random samples with high approximation accuracy, as well as nice theoretic guarantees. Our method shows compelling advantages in performance and accuracy, compared to the state-of-the-art Markov chain Monte Carlo and approximations such as variational Bayes and normalizing flow. We illustrate this approach via several challenging applications, such as sampling from multi-modal distribution, estimating sparse signals in high dimension, and soft-thresholding of a graph with a prior on the degrees.

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arXiv2020

Bayesian Multi-scale Modeling of Factor Matrix without using Partition Tree

The multi-scale factor models are particularly appealing for analyzing matrix- or tensor-valued data, due to their adaptiveness to local geometry and intuitive interpretation. However, the reliance on the binary tree for recursive partitioning creates high complexity in the parameter space, making it extremely challenging to quantify its uncertainty. In this article, we discover an alternative way to generate multi-scale matrix using simple matrix operation: starting from a random matrix with each column having two unique values, its Cholesky whitening transform obeys a recursive partitioning structure. This allows us to consider a generative distribution with large prior support on common multi-scale factor models, and efficient posterior computation via Hamiltonian Monte Carlo. We demonstrate its potential in a multi-scale factor model to find broader regions of interest for human brain connectivity.

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arXiv2019

Tuning-Free Disentanglement via Projection

In representation learning and non-linear dimension reduction, there is a huge interest to learn the 'disentangled' latent variables, where each sub-coordinate almost uniquely controls a facet of the observed data. While many regularization approaches have been proposed on variational autoencoders, heuristic tuning is required to balance between disentanglement and loss in reconstruction accuracy -- due to the unsupervised nature, there is no principled way to find an optimal weight for regularization. Motivated to completely bypass regularization, we consider a projection strategy: modifying the canonical Gaussian encoder, we add a layer of scaling and rotation to the Gaussian mean, such that the marginal correlations among latent sub-coordinates become exactly zero. This achieves a theoretically maximal disentanglement, as guaranteed by zero cross-correlation between one latent sub-coordinate and the observed varying with the rest. Unlike regularizations, the extra projection layer does not impact the flexibility of the previous encoder layers, leading to almost no loss in expressiveness. This approach is simple to implement in practice. Our numerical experiments demonstrate very good performance, with no tuning required.

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arXiv2018

Mixed-Stationary Gaussian Process for Flexible Non-Stationary Modeling of Spatial Outcomes

Gaussian processes (GPs) are commonplace in spatial statistics. Although many non-stationary models have been developed, there is arguably a lack of flexibility compared to equipping each location with its own parameters. However, the latter suffers from intractable computation and can lead to overfitting. Taking the instantaneous stationarity idea, we construct a non-stationary GP with the stationarity parameter individually set at each location. Then we utilize the non-parametric mixture model to reduce the effective number of unique parameters. Different from a simple mixture of independent GPs, the mixture in stationarity allows the components to be spatial correlated, leading to improved prediction efficiency. Theoretical properties are examined and a linearly scalable algorithm is provided. The application is shown through several simulated scenarios as well as the massive spatiotemporally correlated temperature data.

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arXiv2017

Scaling up Data Augmentation MCMC via Calibration

There has been considerable interest in making Bayesian inference more scalable. In big data settings, most literature focuses on reducing the computing time per iteration, with less focused on reducing the number of iterations needed in Markov chain Monte Carlo (MCMC). This article focuses on data augmentation MCMC (DA-MCMC), a widely used technique. DA-MCMC samples tend to become highly autocorrelated in large data samples, due to a miscalibration problem in which conditional posterior distributions given augmented data are too concentrated. This makes it necessary to collect very long MCMC paths to obtain acceptably low MC error. To combat this inefficiency, we propose a family of calibrated data augmentation algorithms, which appropriately adjust the variance of conditional posterior distributions. A Metropolis-Hastings step is used to eliminate bias in the stationary distribution of the resulting sampler. Compared to existing alternatives, this approach can dramatically reduce MC error by reducing autocorrelation and increasing the effective number of DA-MCMC samples per computing time. The approach is simple and applicable to a broad variety of existing data augmentation algorithms, and we focus on three popular models: probit, logistic and Poisson log-linear. Dramatic gains in computational efficiency are shown in applications.

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arXiv2015

Functional Gaussian Process Model for Bayesian Nonparametric Analysis

Gaussian process is a theoretically appealing model for nonparametric analysis, but its computational cumbersomeness hinders its use in large scale and the existing reduced-rank solutions are usually heuristic. In this work, we propose a novel construction of Gaussian process as a projection from fixed discrete frequencies to any continuous location. This leads to a valid stochastic process that has a theoretic support with the reduced rank in the spectral density, as well as a high-speed computing algorithm. Our method provides accurate estimates for the covariance parameters and concise form of predictive distribution for spatial prediction. For non-stationary data, we adopt the mixture framework with a customized spectral dependency structure. This enables clustering based on local stationarity, while maintains the joint Gaussianness. Our work is directly applicable in solving some of the challenges in the spatial data, such as large scale computation, anisotropic covariance, spatio-temporal modeling, etc. We illustrate the uses of the model via simulations and an application on a massive dataset.

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arXiv2014

Joint Hierarchical Gaussian Process Model with Application to Forecast in Medical Monitoring

A novel extrapolation method is proposed for longitudinal forecasting. A hierarchical Gaussian process model is used to combine nonlinear population change and individual memory of the past to make prediction. The prediction error is minimized through the hierarchical design. The method is further extended to joint modeling of continuous measurements and survival events. The baseline hazard, covariate and joint effects are conveniently modeled in this hierarchical structure. The estimation and inference are implemented in fully Bayesian framework using the objective and shrinkage priors. In simulation studies, this model shows robustness in latent estimation, correlation detection and high accuracy in forecasting. The model is illustrated with medical monitoring data from cystic fibrosis (CF) patients. Estimation and forecasts are obtained in the measurement of lung function and records of acute respiratory events. Keyword: Extrapolation, Joint Model, Longitudinal Model, Hierarchical Gaussian Process, Cystic Fibrosis, Medical Monitoring

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