Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
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48 results for “Hilbert-Schmidt metric”
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
A new metric CKCE improves model calibration comparison.
problem Comparing the calibration of probabilistic models is challenging.
method CKCE based on Hilbert-Schmidt norm of conditional mean operators.
result CKCE provides more consistent and robust model calibration comparisons.
We give the twistor description of harmonic maps of the Riemann sphere into the Hilbert-Schmidt Grassmannian. The study of such maps is motivated by the harmonic spheres conjecture formulated in the beginning of this paper.
The Hilbert manifold consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $…
Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.
problem Fixed horizon linear quadratic covariance steering in continuous time with a specific terminal cost.
method Formulates necessary conditions as a coupled matrix ODE two-point boundary value problem, designs a matricial recursive algorithm, and proves convergence.
result Proposes and proves the convergence of a matricial recursive algorithm for solving the steering problem.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
Study optimizes KSD estimation from samples, revealing Hilbert-Schmidt vs trace scales.
problem Optimizing estimation of Kernel Stein Discrepancy from samples.
method Identifying and comparing minimax scales for U-statistic and V-statistic.
result Hilbert-Schmidt norm of Stein covariance operator gives optimal scale.
Optimal transport for functional data using Hilbert-Schmidt operators.
problem Optimal transport for distributions on function spaces with partially represented stochastic maps.
method Regularization technique to restrict transport maps to Hilbert-Schmidt operators, developing an efficient algorithm.
result Existence, uniqueness, and consistency of the Hilbert-Schmidt operator estimate for the transport map.
The aim of this paper is the geometric study of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator. This subgroup of the symplectic group was introduced in Pierre de la Harpe's classical book of Banach-Lie groups. Throughout this paper we will endow the tangent spaces with d…
Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
The paper studies geometric properties of Grassman manifolds within Euclidean spaces.
problem Understanding the geometric structure of Grassman manifolds.
method Analyzing Grassman manifold as a subset of Euclidean space and orthogonal projections.
result Explicit formulas for differential geometry of as a submanifold.
This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.
problem Understanding the Fisher-Rao metric in infinite-dimensional Gaussian settings.
method Explicit description and generalization of finite-dimensional quantities to infinite-dimensional Hilbert spaces.
result The Fisher-Rao metric and related geometric quantities generalize from finite to infinite dimensions.
A Fermionic Grunsky operatormath.CV
New Grunsky operator for disk maps to complex plane.
problem Characterizing domains for Hilbert-Schmidt Grunsky operators.
method Geometric treatment of Smirnov space, pull-back analysis.
result Domains with Hilbert-Schmidt Grunsky operators are Weil-Petersson quasidisks.
Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert space…
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.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
This work improves fair tensor decomposition using a kernel criterion.
problem Learning fair low-rank tensor decompositions with statistical parity.
method Regularizes Canonical Polyadic Decomposition with KHSIC to ensure approximate statistical parity.
result The proposed algorithm achieves better fairness and fit than state-of-the-art FATR.
Discusses MultiFIT for multivariate dependence, comparing it to HSIC tests.
problem Comparing Multiscale Fisher's Independence Test (MultiFIT) to HSIC tests for multivariate dependence.
method Compares MultiFIT to HSIC tests, highlighting exact level control and performance limitations.
result Observes performance limitations of MultiFIT in terms of test power.
We give a Riemannian structure to the set of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold is a universal model for symmetric spaces of the nonc…
In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold.
New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.
problem Applying conditional mean embeddings to complex ML/RL settings with infinite-dimensional RKHSs.
method Developed novel learning rates using interpolation theory for RKHSs, derived explicit adaptive rates for sample estimator.
result Achieved uniform convergence rates in the output RKHS for certain parameter regimes.
Paper proposes a differentially private test for joint dependence among random vectors.
problem Detecting joint dependence among sensitive data while maintaining privacy.
method Differentially private permutation methodology for dHSIC test.
result Proposed test attains minimax optimal power across privacy regimes.
We report on experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced mac…
ICAL improves deep learning model accuracy and NLL with optimized batch labeling.
problem Deep Bayesian Active Learning for efficient model training.
method ICAL uses HSIC to measure dependency and optimizes batch size scaling.
result Significant improvements in model accuracy and NLL on image datasets.
Novel framework for learning infinitesimal generator of stochastic processes.
problem Challenges in learning infinitesimal generator due to unbounded nature and state space dimensionality.
method Introduces a novel framework based on energy functional, integrates physical priors, and uses reduced-rank estimator in RKHS.
result Learning bounds independent of state space dimension and non-spurious spectral estimation.
Kernelized cumulants improve statistical analysis in high-dimensional spaces.
problem Statistical analysis in high-dimensional spaces with low variance estimators.
method Extending cumulants to RKHS using tensor algebra and kernel trick.
result Kernelized cumulants provide new all-purpose statistics with computational tractability.
We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
Robust Independence Tests with Finite Sample Guarantees for Synchronous Stochastic Linear Systemsstat.ML
The paper develops robust tests for detecting independence in synchronous stochastic systems with finite sample guarantees.
problem Detecting independence in synchronous stochastic systems with finite sample guarantees.
method Combines confidence region estimates with permutation tests and dependence measures to detect nonlinear dependence.
result Consistent hypothesis tests for detecting independence under mild assumptions.
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.
For a given Hilbert space , consider the space of self-adjoint projections . In this paper we study the differentiable structure of a canonical sphere bundle over given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \…
The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in …
Kernel-based Joint Independence Tests for Multivariate Stationary and Non-stationary Time Seriesstat.ME
Kernel-based tests detect dependencies in multivariate time series, including stationary and non-stationary data.
problem Detecting dependencies in multivariate time series data, especially non-stationary data.
method Kernel-based statistical tests of joint independence, extending dHSIC to handle both stationary and non-stationary processes.
result Robustly uncovers significant higher-order dependencies in synthetic and real-world data.
GraphITE estimates individual effects of graph-structured treatments.
problem Estimating individual effects of complex treatment structures.
method Graph neural networks and Hilbert-Schmidt Independence Criterion regularization.
result GraphITE outperforms baselines in estimating treatment effects for large numbers of treatments.
Spectral Non-Convex Optimization for Dimension Reduction with Hilbert-Schmidt Independence Criterionstat.ML
The Hilbert Schmidt Independence Criterion (HSIC) is a kernel dependence measure that has applications in various aspects of machine learning. Conveniently, the objectives of different dimensionality reduction applications using HSIC often reduce to the same optimization problem. However, the nonconvexity of the object…
Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistic…
Let be the Banach-Lie group of unitary operators in the Hilbert space which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit of an infinite projection in . This orbit coincides with t…
Measuring conditional independence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connection between conditional independence measures induced by distances on …
A novel disentangled graph autoencoder improves treatment effect estimation from networked observational data.
problem Treatment effect estimation from observational data is challenging due to unconfoundedness assumption and latent confounders.
method Proposes a disentangled variational graph autoencoder to disentangle latent factors and enforce factor independence.
result Extensive experiments show superior performance compared to state-of-the-art approaches.
New method speeds up HSIC for multiple variables.
problem Quadratic computational complexity of HSIC for multiple variables.
method Nyström approximation to HSIC for .
result Consistent Nyström HSIC estimator for .
Framework for generating multiple clusterings from multi-view data.
problem Challenges in finding optimal clustering criteria and handling incomplete multi-view data.
method DiMVMC framework that optimizes multiple decoder deep networks to complete data views and generate shared representations.
result DiMVMC outperforms state-of-the-art competitors in generating multiple clusterings with high diversity and quality.
We investigate the use of a non-parametric independence measure, the Hilbert-Schmidt Independence Criterion (HSIC), as a loss-function for learning robust regression and classification models. This loss-function encourages learning models where the distribution of the residuals between the label and the model predictio…
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…
Study optimizes learning rates for conditional mean embedding estimates.
problem Consistency of kernel ridge regression for conditional mean embedding.
method Adaptive statistical learning rate derived for misspecified setting.
result Upper bound matches optimal rates without assuming finite dimensionality.
Researchers approximate conditional expectation operators using kernel methods.
problem Statistical approximation of conditional expectation operators under minimal assumptions.
method Modifying the domain of the operator, approximating it by Hilbert-Schmidt operators in a reproducing kernel Hilbert space.
result The nonparametric estimate of the operator converges to a specific limiting object.
We investigate the problem of testing whether random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the -dimensional joint …
Kernel dependence measures yield accurate estimates of nonlinear relations between random variables, and they are also endorsed with solid theoretical properties and convergence rates. Besides, the empirical estimates are easy to compute in closed form just involving linear algebra operations. However, they are hampere…