Proposes practical kernel tests for f-divergences with theoretical guarantees.
problem Two-sample testing and machine unlearning evaluation.
method Regularized f-divergence kernel tests, adaptive to hyperparameters. result Different f-divergences highlight localized differences. New divergences help audit DP in high dimensions.
problem Challenges in auditing DP in high-dimensional data.
method Propose kernel Rényi divergence and its regularized version for auditing.
result Regularized kernel Rényi divergence can be estimated from samples in high dimensions.
We study strictly proper scoring rules in the Reproducing Kernel Hilbert Space. We propose a general Kernel Scoring rule and associated Kernel Divergence. We consider conditions under which the Kernel Score is strictly proper. We then demonstrate that the Kernel Score includes the Maximum Mean Discrepancy as a special …
Regularizes f-divergences with MMD to analyze Wasserstein flows.
problem Limitations of f-divergences in measures' support. method Rewriting MMD regularization as Moreau envelope in RKHS, analyzing gradients.
result Analysis of Wasserstein flows of MMD-regularized f-divergences. Paper studies regularized KKL divergence for distributions with disjoint supports.
problem Inability of original KKL divergence to handle distributions with disjoint supports.
method Proposes a regularized variant of KKL divergence, derives bounds, and provides closed-form expression.
result Regularized KKL divergence is well-defined for all distributions and has finite-sample bounds.
Paper bridges VAEs and KDEs for more flexible posterior estimation.
problem Limitations of Gaussian latent space in VAEs and challenges in KL-divergence estimation.
method Approximate posterior with KDEs and derive upper bound of KL-divergence in ELBO.
result Epanechnikov kernel minimizes KL-divergence upper bound asymptotically.
Proposes a new divergence measure for probability distributions.
problem Challenges in estimating divergences from empirical samples.
method Embeds data into RKHS, computes Jensen-Shannon divergence between covariance operators.
result Establishes RJSD as a lower bound on Jensen-Shannon divergence, enabling variational estimation.
The paper develops divergences for Gaussian processes and RKHS settings.
problem Estimating divergences in infinite-dimensional spaces.
method Formulations of Alpha Log-Det divergences, continuity in norm, laws of large numbers, consistent estimation from finite samples.
result Infinite-dimensional divergences can be estimated from finite-dimensional versions with dimension-independent sample complexities.
We extend CS divergence to conditional distributions and show its advantages in time series data and sequential decision making.
problem Quantifying the closeness between conditional distributions.
method Developed and estimated a conditional Cauchy-Schwarz divergence using kernel density estimation.
result Conditional CS divergence outperforms previous methods in time series clustering and sequential decision making.
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
Conditional diffusion models can approximate target distributions well with Gaussian-mixture reverse kernels.
problem Approximating target distributions in conditional diffusion models.
method Using finite Gaussian mixtures with ReLU-network logits as reverse kernels, reducing the problem to static conditional density approximation.
result The resulting neural reverse-kernel class is dense in conditional KL divergence under exact terminal matching.
Proposes a new method for kernel density estimation using stagewise minimization and a simple dictionary.
problem Kernel density estimation with data-adaptive weighting parameters and sparse representation.
method Stagewise minimization algorithm based on U-divergence and a simple dictionary. result Develops non-asymptotic error bound for the proposed estimator.
Recent studies utilize multiple kernel learning to deal with incomplete-data problem. In this study, we introduce new methods that do not only complete multiple incomplete kernel matrices simultaneously, but also allow control of the flexibility of the model by parameterizing the model matrix. By imposing restrictions …
This work improves convergence guarantees for unadjusted HMC in KL and Rényi divergences.
problem Understanding convergence properties of unadjusted HMC in divergences like KL and Rényi.
method One-shot couplings to establish regularization and lift convergence bounds.
result Quantitative control of relative density mismatch and warm-start requirements.
The empirical NTK diverges from the NTK in classification problems during overtraining.
problem The divergence of empirical NTK from NTK in classification problems during overtraining.
method Demonstrated strictly positive definiteness of NTKs for FCNs and ResNets. Proved divergence of neural network parameters during training with cross-entropy loss.
result The empirical NTK does not uniformly converge to the NTK across all times on the training samples as the network width increases.
New method tunes SMC samplers efficiently without high costs.
problem Tuning SMC samplers with unadjusted kernels is challenging.
method Greedy Incremental Divergence Minimization (GIDM) for step size tuning.
result GIDM reduces KL divergence and tunes SMC samplers efficiently.
New method improves neural spike train models by minimizing divergence directly, leading to better performance.
problem Poor performance and divergence issues in spike train models using maximum likelihood estimation.
method Directly minimize maximum mean discrepancy using spike train kernels and stochastic optimization.
result The proposed method generates well-behaved models with better control over feature trade-offs.
Can neural networks learn to compare graphs without feature engineering? In this paper, we show that it is possible to learn representations for graph similarity with neither domain knowledge nor supervision (i.e.\ feature engineering or labeled graphs). We propose Deep Divergence Graph Kernels, an unsupervised method …
Study birth-death dynamics for sampling Gibbs measures with nonconvex potentials.
problem Sampling Gibbs measures with nonconvex potentials.
method Birth-death dynamics, Kullback-Leibler divergence, χ2 divergence, kernel-based approximations, Γ-convergence of gradient flows. result Probability density converges exponentially fast to Gibbs equilibrium measure with a universal rate.
Paper analyzes SVGD algorithm for non-asymptotic convergence.
problem Optimizing a set of particles to approximate a target probability distribution.
method Finite time analysis of SVGD algorithm, providing descent lemma and convergence rates.
result SVGD algorithm decreases the objective at each iteration and converges to the target distribution.
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving Gaussian Process regression efficiency with low-rank approximations.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation.
result Bounds on the divergence and error between exact and approximate GP models.
New recursion formula for non-orientable surfaces resolves divergences.
problem Computing volumes of moduli spaces for non-orientable surfaces.
method Generalization of Mirzakhani's recursion to non-orientable surfaces, handling divergences with integral kernels.
result Regularized volumes can be computed with a cutoff on crosscap size.
Paper proposes a method to stabilize estimation of KL divergence using a discriminator in RKHS.
problem High variance and instability in estimating KL divergence using neural network discriminators.
method Developed a novel construction of the discriminator in RKHS, controlled its complexity, and proved the consistency of the estimator.
result Reduced variance and stabilized training of KL divergence estimates.
Paper develops a method to compare generative models using KL divergence.
problem Lack of principled uncertainty quantification for generative models.
method Employ Kullback-Leibler divergence to measure generative model distance.
result Effective coverage rates and higher power compared to kernel-based methods.
This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infi…
A new method speeds up computation of Sinkhorn divergences to linear time.
problem Expensive computation of Sinkhorn divergences for comparing probability distributions.
method Using positive features to approximate ground costs, reducing computation time to linear.
result Sinkhorn divergences can be computed in linear time, scaling as O(nr).
On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one …
Study characterizes non-collapsed RCD(K, N) spaces using heat kernel metrics.
problem Characterize non-collapsed RCD(K, N) spaces via heat kernel metrics.
method Investigate the second principal term in heat kernel metrics and prove divergence free property.
result Proves non-collapsed property via divergence free property of heat kernel metrics.
Recently, a method called the Mutual Information Neural Estimator (MINE) that uses neural networks has been proposed to estimate mutual information and more generally the Kullback-Leibler (KL) divergence between two distributions. The method uses the Donsker-Varadhan representation to arrive at the estimate of the KL d…
Computing approximate nearest neighbors in high dimensional spaces is a central problem in large-scale data mining with a wide range of applications in machine learning and data science. A popular and effective technique in computing nearest neighbors approximately is the locality-sensitive hashing (LSH) scheme. In thi…
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving the efficiency of Gaussian Process regression while maintaining accuracy.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation, and bounds the divergence and error between exact and approximate models.
result Theoretical bounds on the divergence and error between exact and approximate Gaussian Process models are provided.
We propose a nonparametric statistical test for goodness-of-fit: given a set of samples, the test determines how likely it is that these were generated from a target density function. The measure of goodness-of-fit is a divergence constructed via Stein's method using functions from a Reproducing Kernel Hilbert Space. O…
Gaussian processes improved for ocean current reconstruction and divergence identification.
problem Reconstructing ocean currents from sparse buoy data.
method Proposed a Helmholtz decomposition-based approach to Gaussian processes for better physical modeling.
result Improved inference on ocean currents and divergence identification with minimal computational cost.
Unified deep metric learning approach using neural networks.
problem Learning embeddings of data and extending Euclidean distances.
method Deep Bregman divergences based on neural networks.
result Superior performance on benchmark datasets compared to existing methods.
CO2 algorithm creates coresets for generic smooth divergences efficiently.
problem Efficiently creating coresets for generic smooth divergences.
method CO2 algorithm using functional Taylor expansion and maximum mean discrepancy minimization.
result Poly-logarithmically many data points suffice for Sinkhorn divergence approximation.
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…
Paper studies t-SNE convergence with generalized kernels.
problem Understanding convergence of t-SNE with generalized kernels.
method Concrete formulation of generalized kernels, proving convergence to an equilibrium distribution.
result t-SNE converges to an equilibrium distribution under certain conditions for generalized kernels.
Replacing MSE with f-divergence in diffusion models improves robustness under data contamination.
problem Improving robustness of diffusion models under data contamination.
method Replacing MSE with f-divergence in diffusion models.
result Empirical improvement in performance under data contamination.
Paper presents variational estimates for EBLVMs without structural assumptions.
problem Challenges in learning and evaluating EBLVMs due to intractable true posteriors and partition functions.
method Variational estimates of the score function and its gradient (VaES and VaGES) in a general EBLVM.
result The estimates can be applied to KSD and SM-based methods to learn EBLVMs and estimate Fisher divergence.
Generative Adversarial Networks (GANs) have become a widely popular framework for generative modelling of high-dimensional datasets. However their training is well-known to be difficult. This work presents a rigorous statistical analysis of GANs providing straight-forward explanations for common training pathologies su…
We consider the problem of metric learning subject to a set of constraints on relative-distance comparisons between the data items. Such constraints are meant to reflect side-information that is not expressed directly in the feature vectors of the data items. The relative-distance constraints used in this work are part…
Reduced sample complexity for group-invariant distributions.
problem Improving sample complexity for estimating divergences of group-invariant distributions.
method Quantified reduction in sample complexity for Wasserstein-1 metric and Lipschitz-regularized α-divergences under finite and infinite groups.
result Sample complexity reduction proportional to group size for finite groups, and convergence rate depends on intrinsic dimension for infinite groups.
A new ParVI framework improves particle-based variational inference methods.
problem Non-trivial kernel design in particle-based variational inference methods.
method Proposes a generalized Wasserstein gradient descent (GWG) framework with broader regularizers.
result Demonstrates strong convergence guarantees and effectiveness on simulated and real data.
For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kern…
New method guarantees global convergence in variational inference.
problem Limited convergence to local optima in variational inference.
method Minimizes inclusive KL divergence using neural networks and neural tangent kernel.
result Gradient descent dynamics converge to a unique solution in function space.
Unified technique for sequential estimation of convex divergences.
problem Estimating convex divergences between distributions.
method Martingale methods and maximal inequalities for reverse submartingales.
result Valid time-uniform confidence sequences for arbitrary stopping times.
Empirical Gaussian Processes learn flexible priors from data.
problem Limited effectiveness of standard Gaussian process kernels.
method Estimate mean and covariance functions empirically from data.
result Empirical GPs converge to closest GP to real data generating process.
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…