arXiv research

Study on estimating distances between covariance operators and Gaussian processes.

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…

Extends metrics for SPD matrices to infinite dimensions.

The paper develops robust tests for detecting independence in synchronous stochastic systems with finite sample guarantees.

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. T…

Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.

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 …

New method approximates MMD using pseudo-differential operators and singular values.

A new metric CKCE improves model calibration comparison.

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.

This study approximates distances between Gaussian processes and covariance operators using RKHS.

Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.

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…

Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.

Study optimizes KSD estimation from samples, revealing Hilbert-Schmidt vs trace scales.

The K-sample testing problem involves determining whether K groups of data points are each drawn from the same distribution. Analysis of variance is arguably the most classical method to test mean differences, along with several recent methods to test distributional differences. In this paper, we demonstrate the existe…

Optimal transport for functional data using Hilbert-Schmidt operators.

We investigate the problem of testing whether $d$ 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 $d$-dimensional joint …

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 …

The ability of a human being to extrapolate previously gained knowledge to other domains inspired a new family of methods in machine learning called transfer learning. Transfer learning is often based on the assumption that objects in both target and source domains share some common feature and/or data space. In this p…

The paper studies geometric properties of Grassman manifolds within Euclidean spaces.

New Grunsky operator for disk maps to complex plane.

This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.

Variable selection is of significant importance for classification and regression tasks in machine learning and statistical applications where both predictability and explainability are needed. In this paper, a Copula Entropy (CE) based method for variable selection which use CE based ranks to select variables is propo…

Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.

This work improves fair tensor decomposition using a kernel criterion.

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…

Discusses MultiFIT for multivariate dependence, comparing it to HSIC tests.

Associating genetic markers with a multidimensional phenotype is an important yet challenging problem. In this work, we establish the equivalence between two popular methods: kernel-machine regression (KMR), and kernel distance covariance (KDC). KMR is a semiparametric regression frameworks that models the covariate ef…

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.

Global sensitivity analysis with variance-based measures suffers from several theoretical and practical limitations, since they focus only on the variance of the output and handle multivariate variables in a limited way. In this paper, we introduce a new class of sensitivity indices based on dependence measures which o…

New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.

A statistical test of independence may be constructed using the Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is defined as the distance between the embedding of the joint distribution, and the embedding of the product of the marginals, in a Reproducing Kernel Hilbert Space (RKHS). It has …

Kernelized cumulants improve statistical analysis in high-dimensional spaces.

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.

Meta-learning strategy improves few-shot classification performance.

New research optimizes HSIC estimation rate for translation-invariant kernels.

Kernel-based tests detect dependencies in multivariate time series, including stationary and non-stationary data.

GraphITE estimates individual effects of graph-structured treatments.

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…

Survey of kernels, RKHS, and their applications in machine learning.

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $\{upu^*: u\in U_2({\cal H})\},$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with t…

A novel disentangled graph autoencoder improves treatment effect estimation from networked observational data.

New method speeds up HSIC for multiple variables.

Framework for generating multiple clusterings from multi-view data.

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…