Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.
problem Efficient computation of Sobolev IPM for graph-based probability measures.
method Established relation between Sobolev norm and weighted Lp-norm, proposed novel regularization, leveraged graph structure. result Proposed regularized Sobolev IPM provides closed-form expression for fast computation.
We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure μ. We show that the Sobolev IPM compares two distributions in hig…
Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.
problem Limitation of Le et al. (2025) framework to Lp geometry. method Generalizes Sobolev IPM through Orlicz geometric structure, employing convex functions to capture nuanced geometric relationships.
result GSI-M reduces to a simple univariate optimization problem, achieving remarkable computational efficiency.
We propose the Sobolev Independence Criterion (SIC), an interpretable dependency measure between a high dimensional random variable X and a response variable Y . SIC decomposes to the sum of feature importance scores and hence can be used for nonlinear feature selection. SIC can be seen as a gradient regularized Integr…
Study extends DRO with IPMs, linking robustness to regularization and GANs.
problem Addressing robustness of deep neural networks to adversarial attacks.
method Distributionally Robust Optimization (DRO) with Integral Probability Metrics (IPMs).
result DRO under any IPM corresponds to a family of regularization penalties.
We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on n independent samples from them. Curiously, we show that estimating the IPM itself between probability measures, is not significantly easier than estimating the probabili…
Unified view of KL-divergence and IPMs via DRE, with new DRM metrics.
problem Unified understanding of KL-divergence and IPMs.
method Unified representation via maximum likelihood density-ratio estimation (DRE).
result Unified form of IPMs and novel DRM metrics.
We introduce new families of Integral Probability Metrics (IPM) for training Generative Adversarial Networks (GAN). Our IPMs are based on matching statistics of distributions embedded in a finite dimensional feature space. Mean and covariance feature matching IPMs allow for stable training of GANs, which we will call M…
A new IPM uses ReLU networks to measure probability discrepancies.
problem Measuring the difference between two probability distributions in high dimensions.
method Proposes a new parametric IPM using ReLU neural networks to optimize and distinguish between distributions.
result The proposed IPM has good convergence rates and can be used as a surrogate for other IPMs.
Study minimax rates for density estimation under Huber contamination and Besov IPM losses.
problem Minimax convergence rates of nonparametric density estimation under Huber contamination model with outliers.
method Re-scaled thresholding wavelet series estimator and GAN architectures.
result Achieves minimax optimal convergence rates under Besov IPM losses.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
Proposes tests for comparing high-dimensional manifold samples.
problem Determining if two manifold samples come from the same distribution.
method Integral Probability Metric (IPM) with neural network approximations.
result Tests achieve type-II risk in specific orders of n. New framework improves experimental design using integral probability metrics.
problem Challenges in Bayesian Optimal Experimental Design (BOED) with KL divergence.
method Integrates integral probability metrics (IPMs) for stability and flexibility.
result IPM-based designs yield more robust and accurate credible sets.
The paper explores fair predictors in supervised learning using IPMs and Kolmogorov distance.
problem Achieving fairness in supervised learning with significant demographic effects.
method Identifying conditions for SP-fair predictors and using IPMs to measure unfairness.
result Fair predictors can improve accuracy and are computationally efficient.
New bounds use IPMs to improve generalization in machine learning.
problem Improving generalization bounds in machine learning.
method PAC-Bayes bounds with Integral Probability Metrics (IPM).
result Natural interpolation between worst-case and favorable cases.
IPMs struggle with hyperbolic spaces due to polynomially growing barrier parameters.
problem IPMs' efficiency is hindered in hyperbolic spaces.
method Analyzing the barrier parameter growth in hyperbolic and Hadamard spaces.
result The barrier parameter grows polynomially with the domain's diameter in hyperbolic spaces.
We study the problem of estimating a nonparametric probability density under a large family of losses called Besov IPMs, which include, for example, Lp distances, total variation distance, and generalizations of both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide b…
Improved image generation quality using closed-form discriminator guidance in diffusion models.
problem Enhancing the quality of images generated by diffusion models.
method Theoretical framework to analyze GAN discriminator's effect on Langevin sampling, proposing IPM-GAN optimization as smoothed score-matching.
result Closed-form kernel-based discriminator guidance improves metrics like CLIP-FID and KID.
New error bounds for GANs with nonlinear objective functions derived.
problem Statistical consistency of GANs with nonlinear objective functions.
method Derivation of statistical error bounds for (f,Γ)-GANs using Rademacher complexity. result Proves the statistical consistency of (f,Γ)-GANs. A nonparametric two-sample test using a parametric integral probability metric
problem Detecting distributional differences between two independent samples
method Propose a new two-sample test statistic based on a newly introduced integral probability metric (IPM)
result Establish theoretical guarantees for the associated two-sample testing procedure
The paper proposes a new method for covariate balancing using IPM to improve causal inference.
problem Covariate imbalance in causal inference weighting methods, especially when models are not correctly specified.
method The integral probability metric (IPM) is used to determine optimal weights for treated and control groups.
result The proposed method can be consistent without specifying either the propensity score or outcome regression model.
Develops a new divergence framework that combines f-divergences and IPMs.
problem Comparing distributions that are not absolutely continuous.
method Introduces (f,Γ)-divergences as a two-stage mass-redistribution/mass-transport process. result Improves estimation, learning, and uncertainty quantification in GANs for heavy-tailed distributions.
This paper is concerned with sequential filtering based stochastic optimization (FSO) approaches that leverage a probabilistic perspective to implement the incremental proximity method (IPM). The present FSO methods are derived based on the Kalman filter (KF) and the extended KF (EKF). In contrast with typical methods …
Generative Adversarial Networks (GANs) are powerful models for learning complex distributions. Stable training of GANs has been addressed in many recent works which explore different metrics between distributions. In this paper we introduce Fisher GAN which fits within the Integral Probability Metrics (IPM) framework f…
New divergences improve estimation and GAN training performance.
problem Improving estimation and training in machine learning models.
method Function-space regularized Rényi divergences.
result New divergences reduce variance and improve training performance.
TMLE improves IPM estimation for ecological population dynamics.
problem Estimating key demographic properties from IPM data.
method Targeted Maximum Likelihood Estimation (TMLE) for IPMs.
result Robust and efficient estimators for IPM properties.
We present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball, recovering the original KS test as its simplest case. We give an exact representer…
Paper proposes a new method to quantify uncertainty in machine learning models.
problem Quantifying uncertainty in multiclass classification models.
method Distance-based approach using Integral Probability Metrics (IPMs).
result Effective uncertainty measures for multiclass classification.
The paper bounds the expectation of empirical processes indexed by Hölder classes.
problem Estimating the expectation of the supremum of empirical processes for distributions on bounded sets.
method Providing upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes.
result Deriving non-asymptotic risk bounds for estimating distributions using empirical processes and IPM.
New findings show fixed-kernel discriminators are weaker than feature-learning ones.
problem Comparing performance of fixed-kernel and feature-learning discriminators.
method Using function classes F2 and F1, constructing pairs of distributions, and linking IPMs with sliced Wasserstein distances. result Fixed-kernel IPM and SD cannot discriminate certain distributions that feature-learning IPM and SD can.
Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.
problem Learning high-dimensional distributions with stochastic orders.
method Introduces Choquet-Toland distance and Variational Dominance Criterion, uses input convex maxout networks (ICMNs).
result Proposes surrogates for Choquet-Toland distance and Variational Dominance Criterion with parametric rates.
A new meta-learning framework that assigns weights to source tasks based on target samples.
problem Learning initialization for target tasks with limited labeled examples.
method A general framework that assigns weights to the loss of different source tasks, which can depend on the target samples. Provides upper bounds and develops a learning algorithm based on minimizing the error bound with respect to an empirical IPM.
result Empirically, the weighted meta-learning algorithm finds better initializations than uniformly-weighted meta-learning algorithms.
Paper proposes a new framework to improve policy optimization by aligning real and simulated data distributions.
problem Inaccurate model estimation leads to performance degradation in model-based reinforcement learning.
method Introduces unsupervised model adaptation to minimize the IPM between real and simulated data distributions.
result Achieves state-of-the-art performance in sample efficiency on various continuous control tasks.
The recent empirical success of unsupervised cross-domain mapping algorithms, between two domains that share common characteristics, is not well-supported by theoretical justifications. This lacuna is especially troubling, given the clear ambiguity in such mappings. We work with adversarial training methods based on IP…
New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.
problem Proving Sobolev inequalities on Kähler manifolds.
method Using a classical Bidaut-Véron and Véron approach.
result Proved new Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent works have shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al. suggests a dilemma about GANs' statistical properties: powerful d…
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the W1,2 Sobolev inequality along the Ricci flow …
Proves Sobolev inequality on manifolds with specific curvature properties.
problem Proving Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
method Density and Bakry-Émery Ricci curvature.
result Proves Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.
problem Proving the Michael-Simon-Sobolev inequality for submanifolds of codimension 2.
method Optimal transport techniques.
result Sharpness of the inequality for submanifolds of codimension 2.
Study shows how close functions are to optimal in Riemannian manifolds.
problem Understanding how close functions are to optimal in Riemannian manifolds.
method Analyzes quantitative stability of Sobolev inequalities on compact Riemannian manifolds.
result Functions that nearly saturate a critical Sobolev inequality are quantitatively close to extremal functions.
New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Study continuity of complex Sobolev functions, with applications to Kaehler metrics.
problem Continuity of functions in complex Sobolev spaces.
method Analysis of function regularity in Sobolev spaces, with applications to Kaehler metrics.
result Hermitian generalizations of recent results on Kaehler metrics.