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.
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.
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. …
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.
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.
NPMLE estimator automatically chooses the right model complexity for Gaussian mixtures.
problem Learning mixture models and empirical Bayes estimation with non-convex likelihoods.
method Nonparametric maximum likelihood estimator (NPMLE) using complex-analytic techniques.
result NPMLE solution has O(logn) atoms with high probability, improving model complexity. 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.
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.
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.
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.
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.
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.
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.
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…
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.
The problem of estimating an unknown discrete distribution from its samples is a fundamental tenet of statistical learning. Over the past decade, it attracted significant research effort and has been solved for a variety of divergence measures. Surprisingly, an equally important problem, estimating an unknown Markov ch…
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.
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.
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 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.
Locally private mechanisms' output divergence bounds derived.
problem Bounding divergence between locally private mechanisms' outputs.
method Sharp upper bounds on divergence between input and output distributions.
result Established locally private versions of estimation risk bounds.
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.
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.
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. 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…
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…
We study the Nonparametric Maximum Likelihood Estimator (NPMLE) for estimating Gaussian location mixture densities in d-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accurac…
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.
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.
Bayesian sequence prediction is a simple technique for predicting future symbols sampled from an unknown measure on infinite sequences over a countable alphabet. While strong bounds on the expected cumulative error are known, there are only limited results on the distribution of this error. We prove tight high-probabil…
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. 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…
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
Understanding separation effects on parameter estimation in finite Gaussian mixtures
problem Minimum component separation impact on convergence rates in finite Gaussian mixtures
method Developing a unified geometric framework using Hellinger lower bounds and specialized moment-extraction test functions
result Separation complexity driven by spatial configuration of mixture components
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…
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.
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…
Method identifies low-dimensional structure in high-dimensional probability measures.
problem Identifying low-dimensional structure in high-dimensional probability measures.
method Extends prior work on minimizing majorizations of the Kullback-Leibler divergence to identify optimal approximations within a specific class of measures.
result Connection between dimensional logarithmic Sobolev inequality and approximations with the ansatz.
The study examines conditions for achieving a simple lower bound in estimating mean from samples.
problem Achieving a simple lower bound for estimating the mean of a distribution.
method Analyzes conditions for nearly attaining Le Cam's two-point testing lower bound for mean estimation.
result An algorithm nearly attains the two-point testing rate for mixtures of symmetric, log-concave distributions with a common mean.
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.
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.
Study minimax regret in sequential probability assignment with and without side information.
problem Minimax regret analysis in sequential probability assignment.
method Upper and lower bounds on minimax regret using square-root entropy.
result Lower bound matches upper bound for Donsker classes, up to log factors.
We consider utility maximization problem for semi-martingale models depending on a random factor ξ. We reduce initial maximization problem to the conditional one, given ξ=u, which we solve using dual approach. For HARA utilities we consider information quantities like Kullback-Leibler information and Hellinger inte…
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.
In this paper we extend the work of Smith and Papamichail (1999) and present fast approximate Bayesian algorithms for learning in complex scenarios where at any time frame, the relationships between explanatory state space variables can be described by a Bayesian network that evolve dynamically over time and the observ…