arXiv subjects

XuanLong Nguyen

Publications and source records attributed to XuanLong Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

Learning Mixtures of Nonparametric and Convolutional Measures on Effectively Low-dimensional Affine Spaces

In this paper, we develop a finite mixture of convolutional distributions, a statistical model to analyze continuous data distributed approximately on a mixture of low-dimensional affine subspaces. The observations are assumed independent and identically distributed from the mixture of distributions, where each component arises from a convolution of a distribution supported on a low-dimensional subspace with a suitable noise kernel. We discuss theoretical properties of such class of models, including identifiability under very general conditions - in particular, showing that the minimal representation for such mixtures is uniquely identifiable in a semi-parametric setting. We further study the posterior contraction rates for the parameters for a parametrized class of such models where the supports of the component mixing measures are assumed to be convex polytopes under a suitable well-specified Bayesian regime. This still requires developing novel inverse bounds for problems involving a nested mixture structure, where the mixture kernel is itself another continuous mixture. Our approach for both the identifiability theory and posterior contraction rates is to exploit the geometric structure of the underlying support of the latent measures. Apart from applications in end-member analysis, spectral unmixing and topic models, this study provides a grounded framework for subspace clustering with the goal of exploring conditions for learning multiple latent low-dimensional structures. We illustrate our findings through careful simulation study, which also includes developing new algorithms for such class of models

math.ST↗

arXiv2026

Optimal transport based theory for latent structured models

This article is an exposition on some recent theoretical advances in learning latent structured models, with a primary focus on the fundamental roles that optimal transport distances play in the statistical theory. We aim at what may be the most critical and novel ingredient in this theory: the motivation, formulation, derivation and ramification of inverse bounds, a rich collection of structural inequalities for latent structured models which connect the space of distributions of unobserved structures of interest to the space of distributions for observed data. This theory is illustrated on classical mixture models, as well as the more modern hierarchical models that have been developed in Bayesian statistics, machine learning and related fields.

math.ST↗

arXiv2025

A Bayesian approach to learning mixtures of nonparametric components

Mixture models are widely used in modeling heterogeneous data populations. A standard approach of mixture modeling assumes that the mixture component takes a parametric kernel form. In many applications, making parametric assumptions on the latent subpopulation distributions may be unrealistic, which motivates the need for nonparametric modeling of the mixture components themselves. In this paper, we study finite mixtures with nonparametric mixture components, using a Bayesian nonparametric modeling approach. In particular, it is assumed that the data population is generated according to a finite mixture of latent component distributions, where each component is endowed with a Bayesian nonparametric prior such as the Dirichlet process mixture. We present conditions under which the individual mixture component's distribution can be identified, and establish posterior contraction behavior for the data population's density, as well as densities of the latent mixture components. We develop an efficient MCMC algorithm for posterior inference and demonstrate via simulation studies and real-world data illustrations that it is possible to efficiently learn complex forms of probability distribution for the latent subpopulations. In theory, the posterior contraction rate of the component densities is nearly polynomial, which is a significant improvement over the logarithmic convergence rates of estimating mixing measures via deconvolution.

stat.ME↗

arXiv2025

Dirichlet moment tensors and the correspondence between admixture and mixture of product models

Understanding posterior contraction behavior in Bayesian hierarchical models is of fundamental importance, but progress in this question is relatively sparse in comparison to the theory of density estimation. In this paper, we study two classes of hierarchical models for grouped data, where observations within groups are exchangeable. Using moment tensor decomposition of the distribution of the latent variables, we establish a precise equivalence between the class of Admixture models (such as Latent Dirichlet Allocation) and the class of Mixture of products of multinomial distributions. This correspondence enables us to leverage the result from the latter class of models, which are more well-understood, so as to arrive at the identifiability and posterior contraction rates in both classes under conditions much weaker than in existing literature. For instance, our results shed light on cases where the topics are not linearly independent or the number of topics is misspecified in the admixture setting. Finally, we analyze individual documents' latent allocation performance via the borrowing of strength properties of hierarchical Bayesian modeling. Many illustrations and simulations are provided to support the theory.

math.ST↗

arXiv2024

Learning Topic Hierarchies by Tree-Directed Latent Variable Models

We study a parametric family of latent variable models, namely topic models, equipped with a hierarchical structure among the topic variables. Such models may be viewed as a finite mixture of the latent Dirichlet allocation (LDA) induced distributions, but the LDA components are constrained by a latent hierarchy, specifically a rooted and directed tree structure, which enables the learning of interpretable and latent topic hierarchies of interest. A mathematical framework is developed in order to establish identifiability of the latent topic hierarchy under suitable regularity conditions, and to derive bounds for posterior contraction rates of the model and its parameters. We demonstrate the usefulness of such models and validate its theoretical properties through a careful simulation study and a real data example using the New York Times articles.

math.ST↗

arXiv2024

Dendrogram of mixing measures: Hierarchical clustering and model selection for finite mixture models

We present a new way to summarize and select mixture models via the hierarchical clustering tree (dendrogram) constructed from an overfitted latent mixing measure. Our proposed method bridges agglomerative hierarchical clustering and mixture modeling. The dendrogram's construction is derived from the theory of convergence of the mixing measures, and as a result, we can both consistently select the true number of mixing components and obtain the pointwise optimal convergence rate for parameter estimation from the tree, even when the model parameters are only weakly identifiable. In theory, it explicates the choice of the optimal number of clusters in hierarchical clustering. In practice, the dendrogram reveals more information on the hierarchy of subpopulations compared to traditional ways of summarizing mixture models. Several simulation studies are carried out to support our theory. We also illustrate the methodology with an application to single-cell RNA sequence analysis.

stat.ME↗

arXiv2024

Uncovering Heterogeneity of Solar Flare Mechanism With Mixture Models

The physics of solar flares occurring on the Sun is highly complex and far from fully understood. However, observations show that solar eruptions are associated with the intense kilogauss fields of active regions, where free energies are stored with field-aligned electric currents. With the advent of high-quality data sources such as the Geostationary Operational Environmental Satellites (GOES) and Solar Dynamics Observatory (SDO)/Helioseismic and Magnetic Imager (HMI), recent works on solar flare forecasting have been focusing on data-driven methods. In particular, black box machine learning and deep learning models are increasingly adopted in which underlying data structures are not modeled explicitly. If the active regions indeed follow the same laws of physics, there should be similar patterns shared among them, reflected by the observations. Yet, these black box models currently used in the literature do not explicitly characterize the heterogeneous nature of the solar flare data, within and between active regions. In this paper, we propose two finite mixture models designed to capture the heterogeneous patterns of active regions and their associated solar flare events. With extensive numerical studies, we demonstrate the usefulness of our proposed method for both resolving the sample imbalance issue and modeling the heterogeneity for rare energetic solar flare events.

astro-ph.SR↗

arXiv2023

Modeling random directions of changes in simplex-valued data

We propose models and algorithms for learning about random directions in simplex-valued data. The models are applied to the study of income level proportions and their changes over time in a geostatistical area. There are several notable challenges in the analysis of simplex-valued data: the measurements must respect the simplex constraint and the changes exhibit spatiotemporal smoothness and may be heterogeneous. To that end, we propose Bayesian models that draw from and expand upon building blocks in circular and spatial statistics by exploiting a suitable transformation for the simplex-valued data. Our models also account for spatial correlation across locations in the simplex and the heterogeneous patterns via mixture modeling. We describe some properties of the models and model fitting via MCMC techniques. Our models and methods are applied to an analysis of movements and trends of income categories using the Home Mortgage Disclosure Act data.

stat.ME↗

arXiv2023

Minimum $\Phi$-distance estimators for finite mixing measures

Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the \emph{mixing measure}, a fundamental object that encapsulates all unknown parameters in a mixture distribution. In this paper we propose a general framework for estimating the mixing measure arising in finite mixture models, which we term minimum $\Phi$-distance estimators. We establish a general theory for the minimum $\Phi$-distance estimator, where sharp probability bounds are obtained on the estimation error for the mixing measures in terms of the suprema of the associated empirical processes for a suitably chosen function class $\Phi$. Our framework includes several existing and seemingly distinct estimation methods as special cases using a weakened identifiability condition, but also motivates new estimators. For instance, it extends the minimum Kolmogorov-Smirnov distance estimator to the multivariate setting, and it extends the method of moments to cover a broader family of probability kernels beyond the Gaussian. Moreover, it also includes methods that are applicable to complex (e.g., non-Euclidean) observation domains, using tools from reproducing kernel Hilbert spaces. It will be shown that under general conditions the methods achieve optimal rates of estimation under Wasserstein metrics in either minimax or pointwise sense of convergence; the latter case can be achieved when no upper bound on the finite number of components is given. Also of interest is a sharp inequality that captures the local information geometry for general mixture models precisely in terms of moment differences between mixing measures.

math.ST↗

arXiv2023

On Excess Mass Behavior in Gaussian Mixture Models with Orlicz-Wasserstein Distances

Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.

math.ST↗

arXiv2022

Strong identifiability and parameter learning in regression with heterogeneous response

Mixtures of regression are a powerful class of models for regression learning with respect to a highly uncertain and heterogeneous response variable of interest. In addition to being a rich predictive model for the response given some covariates, the parameters in this model class provide useful information about the heterogeneity in the data population, which is represented by the conditional distributions for the response given the covariates associated with a number of distinct but latent subpopulations. In this paper, we investigate conditions of strong identifiability, rates of convergence for conditional density and parameter estimation, and the Bayesian posterior contraction behavior arising in finite mixture of regression models, under exact-fitted and over-fitted settings and when the number of components is unknown. This theory is applicable to common choices of link functions and families of conditional distributions employed by practitioners. We provide simulation studies and data illustrations, which shed some light on the parameter learning behavior found in several popular regression mixture models reported in the literature.

math.ST↗

arXiv2022

Beyond Black Box Densities: Parameter Learning for the Deviated Components

As we collect additional samples from a data population for which a known density function estimate may have been previously obtained by a black box method, the increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. To model this phenomenon, we consider the \emph{deviating mixture model} $(1-\lambda^{*})h_0 + \lambda^{*} (\sum_{i = 1}^{k} p_{i}^{*} f(x|\theta_{i}^{*}))$, where $h_0$ is a known density function, while the deviated proportion $\lambda^{*}$ and latent mixing measure $G_{*} = \sum_{i = 1}^{k} p_{i}^{*} \delta_{\theta_i^{*}}$ associated with the mixture distribution are unknown. Via a novel notion of distinguishability between the known density $h_{0}$ and the deviated mixture distribution, we establish rates of convergence for the maximum likelihood estimates of $\lambda^{*}$ and $G^{*}$ under Wasserstein metric. Simulation studies are carried out to illustrate the theory.

stat.ML↗

arXiv2021

Modeling Random Directions in 2D Simplex Data

We propose models and algorithms for learning about random directions in two-dimensional simplex data, and apply our methods to the study of income level proportions and their changes over time in a geostatistical area. There are several notable challenges in the analysis of simplex-valued data: the measurements must respect the simplex constraint and the changes exhibit spatiotemporal smoothness while allowing for possible heterogeneous behaviors. To that end, we propose Bayesian models that rely on and expand upon building blocks in circular and spatial statistics by exploiting suitable transformation based on the polar coordinates for circular data. Our models also account for spatial correlation across locations in the simplex and the heterogeneous patterns via mixture modeling. We describe some properties of the models and model fitting via MCMC techniques. Our models and methods are illustrated via a thorough simulation study, and applied to an analysis of movements and trends of income categories using the Home Mortgage Disclosure Act data.

stat.ME↗

arXiv2021

Scalable nonparametric Bayesian learning for heterogeneous and dynamic velocity fields

Analysis of heterogeneous patterns in complex spatio-temporal data finds usage across various domains in applied science and engineering, including training autonomous vehicles to navigate in complex traffic scenarios. Motivated by applications arising in the transportation domain, in this paper we develop a model for learning heterogeneous and dynamic patterns of velocity field data. We draw from basic nonparameric Bayesian modeling elements such as hierarchical Dirichlet process and infinite hidden Markov model, while the smoothness of each homogeneous velocity field element is captured with a Gaussian process prior. Of particular focus is a scalable approximate inference method for the proposed model; this is achieved by employing sequential MAP estimates from the infinite HMM model and an efficient sequential GP posterior computation technique, which is shown to work effectively on simulated data sets. Finally, we demonstrate the effectiveness of our techniques to the NGSIM dataset of complex multi-vehicle interactions.

stat.ML↗

arXiv2021

Functional optimal transport: map estimation and domain adaptation for functional data

We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. { Since probability measures in infinite dimensional spaces generally lack absolute continuity (that is, with respect to non-degenerate Gaussian measures), the Monge map in the standard optimal transport theory for finite dimensional spaces may not exist. Our approach to the optimal transport problem in infinite dimensions is by a suitable regularization technique -- we restrict the class of transport maps to be a Hilbert-Schmidt space of operators.} To this end, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and examine the functional properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.

stat.ML↗

arXiv2020

Functional Aggregate Queries with Additive Inequalities

Motivated by fundamental applications in databases and relational machine learning, we formulate and study the problem of answering functional aggregate queries (FAQ) in which some of the input factors are defined by a collection of additive inequalities between variables. We refer to these queries as FAQ-AI for short. To answer FAQ-AI in the Boolean semiring, we define relaxed tree decompositions and relaxed submodular and fractional hypertree width parameters. We show that an extension of the InsideOut algorithm using Chazelle's geometric data structure for solving the semigroup range search problem can answer Boolean FAQ-AI in time given by these new width parameters. This new algorithm achieves lower complexity than known solutions for FAQ-AI. It also recovers some known results in database query answering. Our second contribution is a relaxation of the set of polymatroids that gives rise to the counting version of the submodular width, denoted by #subw. This new width is sandwiched between the submodular and the fractional hypertree widths. Any FAQ and FAQ-AI over one semiring can be answered in time proportional to #subw and respectively to the relaxed version of #subw. We present three applications of our FAQ-AI framework to relational machine learning: k-means clustering, training linear support vector machines, and training models using non-polynomial loss. These optimization problems can be solved over a database asymptotically faster than computing the join of the database relations.

cs.DB↗

arXiv2020

Robust Unsupervised Learning of Temporal Dynamic Interactions

Robust representation learning of temporal dynamic interactions is an important problem in robotic learning in general and automated unsupervised learning in particular. Temporal dynamic interactions can be described by (multiple) geometric trajectories in a suitable space over which unsupervised learning techniques may be applied to extract useful features from raw and high-dimensional data measurements. Taking a geometric approach to robust representation learning for temporal dynamic interactions, it is necessary to develop suitable metrics and a systematic methodology for comparison and for assessing the stability of an unsupervised learning method with respect to its tuning parameters. Such metrics must account for the (geometric) constraints in the physical world as well as the uncertainty associated with the learned patterns. In this paper we introduce a model-free metric based on the Procrustes distance for robust representation learning of interactions, and an optimal transport based distance metric for comparing between distributions of interaction primitives. These distance metrics can serve as an objective for assessing the stability of an interaction learning algorithm. They are also used for comparing the outcomes produced by different algorithms. Moreover, they may also be adopted as an objective function to obtain clusters and representative interaction primitives. These concepts and techniques will be introduced, along with mathematical properties, while their usefulness will be demonstrated in unsupervised learning of vehicle-to-vechicle interactions extracted from the Safety Pilot database, the world's largest database for connected vehicles.

cs.LG↗

arXiv2020

Convergence of de Finetti's mixing measure in latent structure models for observed exchangeable sequences

Mixtures of product distributions are a powerful device for learning about heterogeneity within data populations. In this class of latent structure models, de Finetti's mixing measure plays the central role for describing the uncertainty about the latent parameters representing heterogeneity. In this paper posterior contraction theorems for de Finetti's mixing measure arising from finite mixtures of product distributions will be established, under the setting the number of exchangeable sequences of observed variables increases while sequence length(s) may be either fixed or varied. The role of both the number of sequences and the sequence lengths will be carefully examined. In order to obtain concrete rates of convergence, a first-order identifiability theory for finite mixture models and a family of sharp inverse bounds for mixtures of product distributions will be developed via a harmonic analysis of such latent structure models. This theory is applicable to broad classes of probability kernels composing the mixture model of product distributions for both continuous and discrete domain $\mathfrak{X}$. Examples of interest include the case the probability kernel is only weakly identifiable in the sense of Ho and Nguyen (2016), the case where the kernel is itself a mixture distribution as in hierarchical models, and the case the kernel may not have a density with respect to a dominating measure on an abstract domain $\mathfrak{X}$ such as Dirichlet processes.

math.ST↗