arXiv subjects

Truyen Nguyen

Publications and source records attributed to Truyen Nguyen.

At least 19 recordsLinked to original sources

arXiv2025

Generalized Sobolev IPM for Graph-Based Measures

We study the Sobolev IPM problem for measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. While Le et al. (2025) achieved scalable computation by relating Sobolev norm to weighted $L^p$-norm, the resulting framework remains intrinsically bound to $L^p$ geometric structure, limiting its ability to incorporate alternative structural priors beyond the $L^p$ geometry paradigm. To overcome this limitation, we propose to generalize Sobolev IPM through the lens of \emph{Orlicz geometric structure}, which employs convex functions to capture nuanced geometric relationships, building upon recent advances in optimal transport theory -- particularly Orlicz-Wasserstein (OW) and generalized Sobolev transport -- that have proven instrumental in advancing machine learning methodologies. This generalization encompasses classical Sobolev IPM as a special case while accommodating diverse geometric priors beyond traditional $L^p$ structure. It however brings up significant computational hurdles that compound those already inherent in Sobolev IPM. To address these challenges, we establish a novel theoretical connection between Orlicz-Sobolev norm and Musielak norm which facilitates a novel regularization for the generalized Sobolev IPM (GSI). By further exploiting the underlying graph structure, we show that GSI with Musielak regularization (GSI-M) reduces to a simple \emph{univariate optimization} problem, achieving remarkably computational efficiency. Empirically, GSI-M is several-order faster than the popular OW in computation, and demonstrates its practical advantages in comparing probability measures on a given graph for document classification and several tasks in topological data analysis.

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arXiv2025

An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph

We investigate optimal transport (OT) for measures on graph metric spaces with different total masses. To mitigate the limitations of traditional $L^p$ geometry, Orlicz-Wasserstein (OW) and generalized Sobolev transport (GST) employ Orlicz geometric structure, leveraging convex functions to capture nuanced geometric relationships and remarkably contribute to advance certain machine learning approaches. However, both OW and GST are restricted to measures with equal total mass, limiting their applicability to real-world scenarios where mass variation is common, and input measures may have noisy supports, or outliers. To address unbalanced measures, OW can either incorporate mass constraints or marginal discrepancy penalization, but this leads to a more complex two-level optimization problem. Additionally, GST provides a scalable yet rigid framework, which poses significant challenges to extend GST to accommodate nonnegative measures. To tackle these challenges, in this work we revisit the entropy partial transport (EPT) problem. By exploiting Caffarelli & McCann (2010)'s insights, we develop a novel variant of EPT endowed with Orlicz geometric structure, called Orlicz-EPT. We establish theoretical background to solve Orlicz-EPT using a binary search algorithmic approach. Especially, by leveraging the dual EPT and the underlying graph structure, we formulate a novel regularization approach that leads to the proposed Orlicz-Sobolev transport (OST). Notably, we demonstrate that OST can be efficiently computed by simply solving a univariate optimization problem, in stark contrast to the intensive computation needed for Orlicz-EPT. Building on this, we derive geometric structures for OST and draw its connections to other transport distances. We empirically illustrate that OST is several-order faster than Orlicz-EPT.

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arXiv2025

Scalable Sobolev IPM for Probability Measures on a Graph

We investigate the Sobolev IPM problem for probability measures supported on a graph metric space. Sobolev IPM is an important instance of integral probability metrics (IPM), and is obtained by constraining a critic function within a unit ball defined by the Sobolev norm. In particular, it has been used to compare probability measures and is crucial for several theoretical works in machine learning. However, to our knowledge, there are no efficient algorithmic approaches to compute Sobolev IPM effectively, which hinders its practical applications. In this work, we establish a relation between Sobolev norm and weighted $L^p$-norm, and leverage it to propose a \emph{novel regularization} for Sobolev IPM. By exploiting the graph structure, we demonstrate that the regularized Sobolev IPM provides a \emph{closed-form} expression for fast computation. This advancement addresses long-standing computational challenges, and paves the way to apply Sobolev IPM for practical applications, even in large-scale settings. Additionally, the regularized Sobolev IPM is negative definite. Utilizing this property, we design positive-definite kernels upon the regularized Sobolev IPM, and provide preliminary evidences of their advantages for comparing probability measures on a given graph for document classification and topological data analysis.

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arXiv2024

Generalized Sobolev Transport for Probability Measures on a Graph

We study the optimal transport (OT) problem for measures supported on a graph metric space. Recently, Le et al. (2022) leverage the graph structure and propose a variant of OT, namely Sobolev transport (ST), which yields a closed-form expression for a fast computation. However, ST is essentially coupled with the $L^p$ geometric structure within its definition which makes it nontrivial to utilize ST for other prior structures. In contrast, the classic OT has the flexibility to adapt to various geometric structures by modifying the underlying cost function. An important instance is the Orlicz-Wasserstein (OW) which moves beyond the $L^p$ structure by leveraging the \emph{Orlicz geometric structure}. Comparing to the usage of standard $p$-order Wasserstein, OW remarkably helps to advance certain machine learning approaches. Nevertheless, OW brings up a new challenge on its computation due to its two-level optimization formulation. In this work, we leverage a specific class of convex functions for Orlicz structure to propose the generalized Sobolev transport (GST). GST encompasses the ST as its special case, and can be utilized for prior structures beyond the $L^p$ geometry. In connection with the OW, we show that one only needs to simply solve a univariate optimization problem to compute the GST, unlike the complex two-level optimization problem in OW. We empirically illustrate that GST is several-order faster than the OW. Moreover, we provide preliminary evidences on the advantages of GST for document classification and for several tasks in topological data analysis.

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arXiv2024

Optimal Transport for Measures with Noisy Tree Metric

We study optimal transport (OT) problem for probability measures supported on a tree metric space. It is known that such OT problem (i.e., tree-Wasserstein (TW)) admits a closed-form expression, but depends fundamentally on the underlying tree structure over supports of input measures. In practice, the given tree structure may be, however, perturbed due to noisy or adversarial measurements. To mitigate this issue, we follow the max-min robust OT approach which considers the maximal possible distances between two input measures over an uncertainty set of tree metrics. In general, this approach is hard to compute, even for measures supported in one-dimensional space, due to its non-convexity and non-smoothness which hinders its practical applications, especially for large-scale settings. In this work, we propose novel uncertainty sets of tree metrics from the lens of edge deletion/addition which covers a diversity of tree structures in an elegant framework. Consequently, by building upon the proposed uncertainty sets, and leveraging the tree structure over supports, we show that the robust OT also admits a closed-form expression for a fast computation as its counterpart standard OT (i.e., TW). Furthermore, we demonstrate that the robust OT satisfies the metric property and is negative definite. We then exploit its negative definiteness to propose positive definite kernels and test them in several simulations on various real-world datasets on document classification and topological data analysis.

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arXiv2023

Scalable Unbalanced Sobolev Transport for Measures on a Graph

Optimal transport (OT) is a popular and powerful tool for comparing probability measures. However, OT suffers a few drawbacks: (i) input measures required to have the same mass, (ii) a high computational complexity, and (iii) indefiniteness which limits its applications on kernel-dependent algorithmic approaches. To tackle issues (ii)--(iii), Le et al. (2022) recently proposed Sobolev transport for measures on a graph having the same total mass by leveraging the graph structure over supports. In this work, we consider measures that may have different total mass and are supported on a graph metric space. To alleviate the disadvantages (i)--(iii) of OT, we propose a novel and scalable approach to extend Sobolev transport for this unbalanced setting where measures may have different total mass. We show that the proposed unbalanced Sobolev transport (UST) admits a closed-form formula for fast computation, and it is also negative definite. Additionally, we derive geometric structures for the UST and establish relations between our UST and other transport distances. We further exploit the negative definiteness to design positive definite kernels and evaluate them on various simulations to illustrate their fast computation and comparable performances against other transport baselines for unbalanced measures on a graph.

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arXiv2023

Dynamic Flows on Curved Space Generated by Labeled Data

The scarcity of labeled data is a long-standing challenge for many machine learning tasks. We propose our gradient flow method to leverage the existing dataset (i.e., source) to generate new samples that are close to the dataset of interest (i.e., target). We lift both datasets to the space of probability distributions on the feature-Gaussian manifold, and then develop a gradient flow method that minimizes the maximum mean discrepancy loss. To perform the gradient flow of distributions on the curved feature-Gaussian space, we unravel the Riemannian structure of the space and compute explicitly the Riemannian gradient of the loss function induced by the optimal transport metric. For practical applications, we also propose a discretized flow, and provide conditional results guaranteeing the global convergence of the flow to the optimum. We illustrate the results of our proposed gradient flow method on several real-world datasets and show our method can improve the accuracy of classification models in transfer learning settings.

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arXiv2022

Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics

Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted $\ell_p$ distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.

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arXiv2021

Adversarial Regression with Doubly Non-negative Weighting Matrices

Many machine learning tasks that involve predicting an output response can be solved by training a weighted regression model. Unfortunately, the predictive power of this type of models may severely deteriorate under low sample sizes or under covariate perturbations. Reweighting the training samples has aroused as an effective mitigation strategy to these problems. In this paper, we propose a novel and coherent scheme for kernel-reweighted regression by reparametrizing the sample weights using a doubly non-negative matrix. When the weighting matrix is confined in an uncertainty set using either the log-determinant divergence or the Bures-Wasserstein distance, we show that the adversarially reweighted estimate can be solved efficiently using first-order methods. Numerical experiments show that our reweighting strategy delivers promising results on numerous datasets.

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arXiv2021

Entropy Partial Transport with Tree Metrics: Theory and Practice

Optimal transport (OT) theory provides powerful tools to compare probability measures. However, OT is limited to nonnegative measures having the same mass, and suffers serious drawbacks about its computation and statistics. This leads to several proposals of regularized variants of OT in the recent literature. In this work, we consider an \textit{entropy partial transport} (EPT) problem for nonnegative measures on a tree having different masses. The EPT is shown to be equivalent to a standard complete OT problem on a one-node extended tree. We derive its dual formulation, then leverage this to propose a novel regularization for EPT which admits fast computation and negative definiteness. To our knowledge, the proposed regularized EPT is the first approach that yields a \textit{closed-form} solution among available variants of unbalanced OT. For practical applications without priori knowledge about the tree structure for measures, we propose tree-sliced variants of the regularized EPT, computed by averaging the regularized EPT between these measures using random tree metrics, built adaptively from support data points. Exploiting the negative definiteness of our regularized EPT, we introduce a positive definite kernel, and evaluate it against other baselines on benchmark tasks such as document classification with word embedding and topological data analysis. In addition, we empirically demonstrate that our regularization also provides effective approximations.

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arXiv2020

On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flow

We consider the maximum mean discrepancy ($\mathrm{MMD}$) GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized $\mathrm{MMD}$ GAN. We show that this flow provides a descent direction minimizing the $\mathrm{MMD}$ on a statistical manifold of probability distributions. We then derive an explicit condition which ensures that gradient descent on the parameter space of the generator in gradient regularized $\mathrm{MMD}$ GAN is globally convergent to the target distribution. Under this condition, we give non asymptotic convergence results of gradient descent in MMD GAN. Another contribution of this paper is the introduction of a dynamic formulation of a regularization of $\mathrm{MMD}$ and demonstrating that the parametric kernelized descent for $\mathrm{MMD}$ is the gradient flow of this functional with respect to the new Riemannian structure. Our obtained theoretical result allows ones to treat gradient flows for quite general functionals and thus has potential applications to other types of variational inferences on a statistical manifold beyond GANs. Finally, numerical experiments suggest that our parametric kernelized gradient flow stabilizes GAN training and guarantees convergence.

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arXiv2018

Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.

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arXiv2018

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

We study regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\A$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\calM^{q,λ}$ or weighted space $L^q_w$ whenever $|\F|^{\frac{1}{p-1}}$ is respectively in $\calM^{q,λ}$ or $L^q_w$, where $q$ is any number greater than $p$ and $w$ is any weight in the Muckenhoupt class $A_{\frac{q}{p}}$. In addition, our two-weight estimate allows the possibility to acquire the regularity for $\nabla u$ in a weighted Morrey space that is different from the functional space that the data $|\F|^{\frac{1}{p-1}}$ belongs to.

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arXiv2017

Interior Calderón-Zygmund estimates for solutions to general parabolic equations of $p$-Laplacian type

We study general parabolic equations of the form $u_t = div A(x,t, u,D u) + div(|F|^{p-2} F)+ f$ whose principal part depends on the solution itself. The vector field $A$ is assumed to have small mean oscillation in $x$, measurable in $t$, Lipschitz continuous in $u$, and its growth in $Du$ is like the $p$-Laplace operator. We establish interior Calderón-Zygmund estimates for locally bounded weak solutions to the equations when $p>2n/(n+2)$. This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto \cite{D2} and the maximal function free approach by Acerbi and Mingione \cite{AM}.

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arXiv2016

Global $W^{1,p}$ estimates for solutions to the linearized Monge--Ampère equations

In this paper, we investigate regularity for solutions to the linearized Monge-Ampère equations when the nonhomogeneous term has low integrability. We establish global $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the equations with right hand side in $L^q$ where $n/2<q\leq n$. These estimates hold under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{1,p}$ estimates of N. Winter for fully nonlinear, uniformly elliptic equations.

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arXiv2015

Interior gradient estimates for quasilinear elliptic equations

We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F} $ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1 p$, and we show that similar results also hold true in the setting of {\it Orlicz} spaces. Our regularity estimates extend results which are only known for the case $\mathbf{A}$ is independent of $u$ and they complement the well-known interior $C^{1,α}$- estimates obtained by DiBenedetto \cite{D} and Tolksdorf \cite{T} for general quasilinear elliptic equations.

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arXiv2015

Local gradient estimates for degenerate elliptic equations

This paper is focused on the local interior $W^{1,\infty}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u, \nabla u)] +b(x, u, \nabla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^\infty$-norm for the solution's gradient in terms of its local $L^p$-norm. Specifically, we prove \begin{equation*} \|\nabla u\|_{L^\infty(B_{\frac{R}{2}}(x_0))}^p \leq \frac{C}{|B_R(x_0)|}\int_{B_R(x_0)}|\nabla u(x)|^p dx. \end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.

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arXiv2013

Non-existence and Non-uniqueness for Multidimensional Sticky Particle Systems

The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.

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