The study explores generalized divergences and exponential families with a focus on sufficient conditions and laws of large numbers.
problem Generalization of Kullback-Leibler divergence and exponential families.
method Investigation of (h,τ)-divergence and (h,τ)-exponential families, definition of (h,τ)-dependence, proof of law of large numbers. result Sufficient condition for (h,τ)-divergence to induce Hessian structure on (h,τ)-exponential family, proof of law of large numbers. Study explores relationship between Hölder and FDPD divergences.
problem Understanding the relationship between Hölder and FDPD divergences.
method Intersection and generalization of divergence families, proving nonnegativity, deriving inequalities.
result Established ξ-Hölder divergence and derived inequalities. 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…
Paper studies statistical manifolds with logarithmic divergences.
problem Understanding statistical manifolds induced by logarithmic divergences.
method Constructs dual foliation of the statistical manifold.
result Extends dual foliation of a dually flat manifold.
Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, th…
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…
EM algorithm converges in KL divergence for exponential families via mirror descent.
problem Lack of understanding of EM's non-asymptotic convergence properties.
method Viewing EM as a mirror descent algorithm, showing convergence rates in KL divergence.
result KL divergence rates for EM in exponential families, invariant to parametrization.
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 f-divergence measures improve robustness in noisy label learning.
problem Improving robustness in learning with noisy labels.
method Derived decoupling property of f-divergence measures under label noise. result Properly defined f-divergence measures are robust with label noise. We propose a Laplace approximation that creates a stochastic unit from any smooth monotonic activation function, using only Gaussian noise. This paper investigates the application of this stochastic approximation in training a family of Restricted Boltzmann Machines (RBM) that are closely linked to Bregman divergences.…
ERM with f-divergence regularization yields unique solution.
problem Optimizing empirical risk with f-divergence. method Mild conditions on f lead to unique optimal measure. result Equivalence of ERM-fDR to different f-divergence regularization. Study on divergence and thickness for Coxeter groups, generalizing previous work.
problem Characterizing and bounding divergence and thickness for Coxeter groups.
method Characterization of linear divergence, introduction of hypergraph index, new construction of Coxeter systems.
result Upper bounds on divergence and thickness for Coxeter groups, conjectured to be equalities.
Generalizes bias-variance decomposition for Bregman divergences.
problem No specific problem stated; generalization of bias-variance for Bregman divergences.
method Provided a generalization of the bias-variance decomposition for Bregman divergences.
result A clear, standalone derivation of the bias-variance decomposition for Bregman divergences.
New separation concepts for Anosov representations help bound Thurston asymmetric metric.
problem Understanding diverging families of Anosov representations.
method Introducing separation concepts and analyzing combinatorial invariants.
result Critical exponent asymptotic to a graph invariant.
We describe the underlying probabilistic interpretation of alpha and beta divergences. We first show that beta divergences are inherently tied to Tweedie distributions, a particular type of exponential family, known as exponential dispersion models. Starting from the variance function of a Tweedie model, we outline how…
New variational formula for Rényi divergences improves neural network estimation in high dimensions.
problem Estimating Rényi divergences in high-dimensional systems.
method Derive and apply a variational formula for Rényi divergences over various function spaces.
result Neural network estimators of Rényi divergences are consistent under certain conditions.
A new family of multi-distribution divergences is characterized for fairness and other problems.
problem Comparing more than two probability distributions in a consistent way.
method Characterized a new family of multi-way coincidence divergences.
result The new family of multi-distribution Rényi divergences is necessary and arises from several independent routes.
Black box variational inference (BBVI) with reparameterization gradients triggered the exploration of divergence measures other than the Kullback-Leibler (KL) divergence, such as alpha divergences. In this paper, we view BBVI with generalized divergences as a form of estimating the marginal likelihood via biased import…
The paper improves guarantees for VI in symmetric cases.
problem Approximating intractable densities via VI with misspecified families.
method Extends previous robust VI results to wider divergences and non-log-concave targets.
result Guarantees for exact recovery of target mean and correlation matrix under various conditions.
New divergences improve score-based methods for multi-modal distributions.
problem Blindness problem in score-based divergences for multi-modal distributions.
method Proposed a new family of divergences to mitigate blindness.
result Improved performance in density estimation compared to traditional approaches.
This work broadens calibeating to various proper losses using Bregman divergence.
problem Calibration for a wide range of proper losses.
method Regret minimization and Bregman divergence approach.
result U-calibration results for a family of Tsallis losses with logarithmic regret and dimension independence.
This manuscript develops the theory of agglomerative clustering with Bregman divergences. Geometric smoothing techniques are developed to deal with degenerate clusters. To allow for cluster models based on exponential families with overcomplete representations, Bregman divergences are developed for nondifferentiable co…
Study optimal transport costs with zero MTW tensor, finding new families of costs and divergence functions.
problem Characterize optimal transport costs with zero MTW tensor.
method Optimal transport theory, information geometry, solving nonlinear ODEs.
result Found new families of costs and divergence functions.
This work generalizes calibeating for a broader range of proper losses using Bregman divergence.
problem Calibration for a wide range of proper losses beyond Brier and log loss.
method Regret minimization based on Bregman divergence for a family of proper losses.
result U-calibration results for a family of Tsallis losses with logarithmic regret and dimension independence.
New clustering method for exponential family data.
problem Improving clustering for non-Gaussian data.
method Bregman Power k-Means algorithm.
result Outperforms existing methods in non-Gaussian data settings.
Innovative inequalities for divergences with applications in PAC-Bayesian bounds and Monte Carlo.
problem Developing new inequalities for divergences.
method Introducing novel change of measure inequalities for f-divergences and α-divergences. result Applications in PAC-Bayesian bounds and Monte Carlo estimates.
New principle controls graph-informed adversarial discrepancies.
problem Graph-informed adversarial learning for interpolative divergences.
method Proves infimal subadditivity for interpolative divergences.
result Graph-informed adversarial learning is justified for interpolative divergences.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
Study of generalized Csiszár divergences and their application to Cramér-Rao bounds.
problem Deriving lower bounds for estimator variance using generalized divergences.
method Applied Eguchi's theory to derive Fisher information metric and dual affine connections.
result More widely applicable Cramér-Rao inequality for escort distributions.
We review recent results about the maximal values of the Kullback-Leibler information divergence from statistical models defined by neural networks, including naive Bayes models, restricted Boltzmann machines, deep belief networks, and various classes of exponential families. We illustrate approaches to compute the max…
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
New metrics defined on SPD matrices link to divergences and curvature.
problem Defining and characterizing metrics on SPD matrices.
method Developed a principle of deformed metrics and introduced balanced bilinear forms.
result Introduce Mixed-Euclidean metrics with negative sectional curvature.
All you need is log
problem Comparing multiple probability distributions
method Characterizing multi-distribution Rényi divergences
result Found a canonical multi-distribution Rényi calculus
New algorithm optimizes unimodal bandits using empirical divergence.
problem Optimizing decisions in multi-armed bandit problems with unimodal distributions.
method Indexed Minimum Empirical Divergence (IMED) adapted for unimodal structure.
result IMED-UB algorithm optimally exploits unimodal structure.
This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared Euclidean distance, Kullback-Leibler and Itakura-Saito divergences as special cases. Learning the model order is impo…
Paper introduces f-divergence variational inference for broader application.
problem Variational inference limited to specific divergences.
method Generalizes variational inference to all f-divergences using f-divergence minimization.
result Unified framework for variational inference with arbitrary f-divergences.
This work extends alpha-beta divergences to complex data and finds closed-form solutions.
problem Approximating complex random vectors.
method Extending alpha-beta divergences to complex data and optimizing the alpha-beta mean distortion.
result Closed-form expression for the centroid of complex random vectors.
Estimates KL divergence with fairness considerations for sub-populations.
problem Fairly estimate KL divergence between distributions considering sub-populations.
method Proposes multi-group attribution for KL divergence estimation, derived from multi-calibration.
result Shows multi-group attribution provides better KL divergence estimates conditioned on sub-populations.
A theoretical framework for non-negative matrix factorization based on generalized dual Kullback-Leibler divergence, which includes members of the exponential family of models, is proposed. A family of algorithms is developed using this framework and its convergence proven using the Expectation-Maximization algorithm. …
VI struggles to fully quantify uncertainty when distributions don't factorize.
problem Uncertainty quantification in non-factorizable distributions.
method Analysis of variational inference trade-offs and divergence choices.
result Different divergences yield different measures of uncertainty in VI.
New model learns better policies from expert demonstrations with higher efficiency.
problem Learning accurate policies from expert demonstrations with high efficiency.
method Generative adversarial imitation learning (GAIL) model that learns f-divergence automatically. result Learns better policies with higher data efficiency in physics-based control tasks.
Proves uniqueness of certain S1-symmetric gravitational instantons.
problem Proving uniqueness of S1-symmetric gravitational instantons. method Using a divergence identity and results from the G-signature theorem. result Proof of the S1-symmetric Euclidean Black Hole Uniqueness conjecture. New measures generalize existing ones, linking information and risk.
problem Linking information measures and risk in statistical decision problems.
method Introducing new families of divergence measures and deriving an information processing equality.
result Extension of variational φ-divergence representation to multiple distributions. The paper proves learning-curve monotonicity for maximum likelihood estimators in various parametric settings.
problem Establishing monotonicity guarantees for maximum likelihood estimators.
method Variants of GPT-5.2 Pro were used to derive the results.
result The paper proves monotonicity for maximum likelihood estimators in Gaussian and Gamma variables.
This paper introduces a new method to train normalizing flows using precision-recall divergences.
problem Training generative models with mode dropping and low-quality samples.
method Introduces PR-divergences and proposes a novel generative model to minimize precision-recall trade-offs.
result Normalizing flows can be trained to achieve specific precision-recall trade-offs using PR-divergences.
New geometry for optimal transport cost based on Bregman divergences.
problem Optimal transport cost calculation with Bregman divergences.
method Established properties, defined interpolations, constructed dualistic geometry.
result Derived generalized Pythagorean inequality and Bregman-Wasserstein barycenters.
New analysis of annealing paths in sampling and estimation.
problem Sampling from complex distributions and estimating normalization constants.
method Extending known results on Bregman divergence to quasi-arithmetic means under monotonic embedding.
result Analogous result for quasi-arithmetic means, highlighting the interplay between means, parametric families, and divergence functionals.
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.