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 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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
Let U2(H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit {upu∗:u∈U2(H)}, of an infinite projection p in H. This orbit coincides with t…
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 O(logn/n) rates without assuming finite dimensionality. 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…
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…
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.
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.
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…
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.
The Hilbert manifold Σ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits Ω⊂Σ 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.
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.
problem Learning from non-independent and non-identically distributed data.
method Data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space.
result Achieved novel risk bounds for covariance operator estimation and operator learning.
We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning prob- lems with nonscalar outputs like multi-task learning and structured out- put prediction. We show that multi-task ke…
We offer a new, rigorous approach to conditional mean embeddings without operator constraints.
problem Lack of rigorous, operator-free approach to conditional mean embeddings.
method Measure-theoretic approach to conditional mean embeddings.
result Natural regression interpretation and universal consistency of empirical estimates.
The paper studies geometric properties of Grassman manifolds within Euclidean spaces.
problem Understanding the geometric structure of Grassman manifolds.
method Analyzing Grassman manifold G(E) as a subset of Euclidean space E and orthogonal projections. result Explicit formulas for differential geometry of G(E) as a submanifold. We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the nonc…
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…
This study approximates distances between Gaussian processes and covariance operators using RKHS.
problem Approximating distances between Gaussian processes and covariance operators from finite samples.
method Using reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators, the study shows how to consistently and efficiently estimate Sinkhorn divergence from finite samples.
result Convergence rates are dimension-independent and of the same order as Hilbert-Schmidt distance.
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.
We study the forward price dynamics in commodity markets realized as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović. The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite dimensional Lévy proces…
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.
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.
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…
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.
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…
HOPE uses Hilbert space to deconstruct deep network representations.
problem Deconstructing learned representations in deep networks is challenging.
method Introduces Hilbert Operator for Progressive Encoding (HOPE) to deconstruct network weights.
result HOPE provides an unbiased approach to network compression and fine-tuning.
The article analyzes LCE in Hilbert space, deriving new formulas and regularisation methods.
problem Analyzing conditional expectation in infinite-dimensional Hilbert space.
method Establishing analytical properties and regularisation for LCE in Hilbert space, deriving new formulas.
result Simple derivation and intuitive justification of conditional mean embedding formula.
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.
This paper deals with the problem of nonparametric independence testing, a fundamental decision-theoretic problem that asks if two arbitrary (possibly multivariate) random variables X,Y are independent or not, a question that comes up in many fields like causality and neuroscience. While quantities like correlation o…
Paper introduces a new measure of conditional dependence avoiding matrix inversions.
problem Measuring conditional dependence between two phenomena influenced by a confounder.
method Uses U-statistics pruning to avoid matrix inversions and re-interpret independence.
result Proposes a novel measure of conditional dependence that avoids matrix inversions.
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.
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.
For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper we study the differentiable structure of a canonical sphere bundle over P(H) given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \…
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 …
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.