New f-Betas for portfolio optimization using f-divergence risk measures.
problem Optimizing portfolio performance under varying market conditions.
method Derive f-Betas and Hellinger-Betas, using f-divergence risk measures.
result Demonstrated new Beta metrics provide better performance under stress.
Formula derived for curvature in measure spaces.
problem Deriving sectional curvature in measure spaces.
method Explicit formula derivation for sectional curvature in M(M) with metrics HK and W2. result Curvature analysis in M(M) reveals both negative and positive components. Private minimum Hellinger distance estimators maintain robustness and efficiency while ensuring privacy.
problem Ensuring privacy in robust statistical estimation.
method Derive private minimum Hellinger distance estimators satisfying Hellinger differential privacy.
result Private minimum Hellinger distance estimators retain robustness and efficiency under privacy constraints.
This paper uses UOT metrics for better dimensionality reduction and classification/clustering.
problem Improving dimensionality reduction and classification/clustering methods.
method Uses Hellinger--Kantorovich metric from unbalanced optimal transport (UOT).
result UOT outperforms Euclidean and OT-based methods in classification and clustering tasks.
A new algorithm for better decision-making in recommendation systems.
problem Stochastic multi-armed bandit problem and cold start problem in recommender systems.
method Proposes Hellinger-UCB, a variant of UCB algorithm using squared Hellinger distance.
result Hellinger-UCB reaches the theoretical lower bound and outperforms other algorithms in practical applications.
Robust test for distributions under Hellinger distance, simpler than optimal tests.
problem Testing and estimating distributions robustly under Hellinger distance.
method Simple robust hypothesis test with optimal sample complexity, robust to Hellinger distance perturbations.
result Empirically demonstrated robustness and power of the test on canonical distributions.
Sharp inequality between TV and Hellinger distances for Gaussian mixtures.
problem Understanding the relationship between total variation and Hellinger distances for Gaussian mixtures.
method Established a general upper bound on Hellinger distance in terms of TV distance raised to a power, demonstrating sharpness with specific examples.
result The Hellinger distance between two Gaussian mixtures is bounded by the TV distance raised to a power 1−o(1), where o(1) is of order 1/loglog(1/TV). Study robust hypothesis testing under Hellinger distance, proving lower bounds and providing tests.
problem Testing close variants of specified distributions robustly to Hellinger distance.
method Lower bound on slack factor, testing with Hellinger balls, symmetric chi-squared distance analysis.
result Lower bound on slack factor quantifies robustness under misspecification.
Constructs portfolios based on Hellinger distance to normal, finding market invariance.
problem Finding a market invariant for portfolio construction.
method Uses Hellinger distance to normal distribution for portfolio construction and analysis.
result Minimum Hellinger distance varies drastically between markets, suggesting market invariance.
Proposes new loss functions for GANs to improve estimation accuracy and robustness.
problem Improving the training of GANs to achieve more accurate and robust models.
method Introduces Hellinger-type loss functions and analyzes their statistical properties.
result Demonstrates improved estimation accuracy and robustness of the proposed loss functions.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
Enhances SDR via Hellinger correlation for better data dependency understanding.
problem Improving sufficient dimension reduction in single-index models.
method Developed a new method using Hellinger correlation for detecting the dimension reduction subspace.
result Significantly enhances and outperforms existing SDR methods through deeper data dependency understanding.
Paper derives convergence rates for NPMLE in Hellinger distance using deep neural networks.
problem Difficulty in proving convergence of excess risk in nonparametric logistic regression.
method Unified approach for analyzing NPMLE, deriving convergence rates in Hellinger distance.
result Derives nearly optimal convergence rates for NPMLE with deep neural networks.
The study analyzes the evolution of Gaussian measures under a specific gradient flow.
problem Analyzing the evolution of Gaussian measures under a specific gradient flow.
method Derives ordinary differential equations governing the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow.
result Exponential convergence to equilibrium demonstrated through Polyak-Lojasiewicz-type inequalities.
Unified framework for model-based RL with sample complexity guarantees.
problem Designing efficient posterior sampling methods for model-based RL.
method Optimistic posterior sampling, Hellinger distance reduction, data likelihood measurement.
result Unified algorithms with state-of-the-art sample complexity guarantees.
We study likelihood-based methods for distribution regression with deep generative models.
problem Distribution regression with high-dimensional responses concentrated on a low-dimensional manifold.
method Likelihood-based approach using conditional deep generative models.
result Convergence rates for estimating conditional distributions in Hellinger and Wasserstein metrics.
Classifiers trained on data sets possessing an imbalanced class distribution are known to exhibit poor generalisation performance. This is known as the imbalanced learning problem. The problem becomes particularly acute when we consider incremental classifiers operating on imbalanced data streams, especially when the l…
Develops an online Gaussian process method that maintains convergence guarantees without sample complexity issues.
problem The computational intractability of Gaussian processes with streaming data.
method Parsimonious Online Gaussian Processes (POG) that maintains asymptotic consistency with bounded memory.
result POG preserves convergence guarantees to the population posterior with finite memory, even for constant error radius.
New method improves learning from multiple correlated data trajectories.
problem Learning from multiple correlated data trajectories without mixing assumptions.
method Hellinger localization framework for maximum likelihood estimation.
result Instance-optimal bounds that scale with full data budget under broad conditions.
A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.
problem Efficiently retrieving patterns from large datasets of weighted point clouds.
method Derived retrieval dynamics as a SHK gradient flow, discretized for a deterministic algorithm.
result Proved basin invariance, geometric convergence, and robust recovery from perturbations.
The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept sim…
SLERP interpolation optimizes dynamic weight rebalancing in AMMs.
problem Optimizing dynamic weight rebalancing in automated market makers (AMMs).
method Riemannian geometry and SLERP interpolation.
result SLERP interpolation minimizes the KL divergence loss in dynamic weight rebalancing.
SQFA learns features maximizing Fisher-Rao distance for better classification.
problem Improving classification accuracy through feature learning.
method SQFA learns linear features maximizing Fisher-Rao distance between class-conditional distributions.
result SQFA-H features achieve the best classification accuracy.
In reinforcement learning (RL), temporal abstraction still remains as an important and unsolved problem. The options framework provided clues to temporal abstraction in the RL, and the option-critic architecture elegantly solved the two problems of finding options and learning RL agents in an end-to-end manner. However…
Boosting variational inference (BVI) approximates an intractable probability density by iteratively building up a mixture of simple component distributions one at a time, using techniques from sparse convex optimization to provide both computational scalability and approximation error guarantees. But the guarantees hav…
We present new excess risk bounds for general unbounded loss functions including log loss and squared loss, where the distribution of the losses may be heavy-tailed. The bounds hold for general estimators, but they are optimized when applied to η-generalized Bayesian, MDL, and empirical risk minimization estimators. …
We show that the square Hellinger distance between two Bayesian networks on the same directed graph, G, is subadditive with respect to the neighborhoods of G. Namely, if P and Q are the probability distributions defined by two Bayesian networks on the same DAG, our inequality states that the square Hellinger di…
This work develops sampling methods for differential privacy using SHK geometry.
problem Approximating sampling for the exponential mechanism in differential privacy.
method Develops perturbation theory for SHK gradient flows and applies to differential privacy.
result Derives time-dependent Pure-DP guarantees and Approximate-DP certificates.
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…
This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded d-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. Furthermore, the optimal portfolios for each of…
Paper proposes a new method for density estimation using squared Hellinger distance.
problem Density estimation using moment methods is sensitive to the choice of functions.
method Proposes a non-classical parametrization using squared Hellinger distance for density estimation.
result The proposed method does not require choosing functions and can be solved by convex optimization.
We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev H˙1-metric. We construct an explicit isometry f…
In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used…
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
Geometric method captures rare topics and temporal alignment in co-author networks.
problem Missing rare topics and smooth temporal alignment in topic modeling.
method Integrates multimodal text and co-author network data using Hellinger distances and Ward's linkage.
result Effective identification of rare topics and visualization of topic drift over time.
Formula derived for sample complexity in binary hypothesis testing.
problem Determine the minimum number of samples to distinguish between two distributions.
method Developed a formula for sample complexity in both prior-free and Bayesian settings, using Jensen-Shannon and Hellinger divergences.
result Formula characterizes sample complexity for a wide range of error parameters, up to multiplicative constants.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family o…
Proposes a continuous relaxation for discrete Bayesian optimization.
problem Efficiently optimizing discrete data with limited target observations.
method Continuous relaxation of objective function, incorporating prior knowledge.
result Optimization can be computationally tractable with few observations.
Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so through pairwise kernel evaluations between pdfs (or sample sets). While such an appr…
The Social Internet of Things (SIoT), integration of the Internet of Things and Social Networks paradigms, has been introduced to build a network of smart nodes that are capable of establishing social links. In order to deal with misbehaving service provider nodes, service requestor nodes must evaluate their trustworth…
Paper constructs a CRRIX index to assess cryptocurrency market risks from regulatory changes.
problem Lack of indices quantifying regulatory risks in cryptocurrencies.
method CRRIX index based on news coverage frequency, using Latent Dirichlet Allocation and Hellinger distance.
result CRRIX successfully captures major policy-changing moments and synchronizes with market volatility.
Study nonparametric density estimation via measure transport, achieving optimal rates.
problem Nonparametric density estimation with optimal rates.
method Measure transport, penalized maximum likelihood, and sieved wavelet estimators.
result Achieve minimax optimal convergence rates over Hölder classes of densities.
Paper resolves open problems on sample complexity in binary hypothesis testing.
problem Open problems in distributed simple binary hypothesis testing under information constraints.
method One-shot lower bound on Bayes error, streamlined sample complexity formula, reverse data-processing inequality.
result Optimally tight sample complexity bounds for communication-constrained simple binary hypothesis testing.
Unified framework for deriving generalization bounds in supervised learning.
problem Generalization error bounds in supervised learning.
method Data Processing Inequality PAC-Bayesian framework.
result Unified bounds on binary Kullback-Leibler generalization gap for various divergences.
Enhanced Gaussian process models accelerate optimization and posterior approximation.
problem Improving the accuracy and speed of Gaussian process models for optimization and inference.
method Introduces a random exploration step to classical GP-UCB algorithms, facilitating faster convergence.
result New algorithms achieve nearly optimal convergence rates and provide bounds for Hellinger distance.
Estimates score function from data with optimal rate in high dimensions.
problem Estimating the score function of an unknown probability distribution from data.
method Empirical Bayes smoothing with a Gaussian kernel.
result Optimal rate of estimation ildeΘ(n−d+42) for d dimensions. New framework estimates staged tree models using hierarchical clustering on the probability simplex.
problem Estimating staged tree models with context-specific dependencies.
method Hierarchical clustering on the probability simplex, using simplex-based divergences and linkage methods.
result Total Variation divergence with Ward.D2 linkage produces staged trees with better model fit, structure recovery, and computational efficiency.