SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
Invariant covariant derivatives on homogeneous spaces are characterized.
problem Understanding invariant covariant derivatives on homogeneous spaces.
method Expressing covariant derivatives in terms of horizontally lifted vector fields and bilinear maps.
result Existence and characterization of invariant covariant derivatives.
Classification of SL(n) covariant valuations on Orlicz spaces.
problem Classifying continuous SL(n) covariant valuations on Orlicz spaces.
method Complete classification without symmetric assumptions, focusing on moment matrix and a new functional in dimension two.
result The moment matrix is the only SL(n) covariant valuation for n≥3, and a new functional appears in dimension two.
The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w …
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form…
Symplectic forms from two phase spaces are proven equivalent.
problem Equivalence of symplectic forms from different phase spaces.
method Proof of equivalence for theories over space-time with boundary.
result Symplectic forms derived from canonical and covariant phase spaces are equivalent.
Algorithm improves SVM classification in non-Euclidean spaces.
problem Limitations of traditional SVM in non-Euclidean spaces.
method Covariance-adjusted SVM using Cholesky Decomposition.
result Cholesky-SVM outperforms traditional SVM in non-Euclidean spaces.
Introduces intrinsic Riemannian cross-covariance for manifold-valued random objects.
problem Covariance estimation for random objects on Riemannian manifolds.
method Defines covariance and correlation via parallel transport.
result Proposed covariance is independent of coordinate choices.
Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.
problem Classifying SL(n) covariant matrix-valued valuations on Lp-spaces.
method Established a complete classification for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx), eliminating matrix symmetry assumption.
result Unique characterization of such valuations by the moment matrix in n>2, rotation matrix in 2D.
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.
Constructs covariant derivatives for Ehresmann connections.
problem Developing a method for covariant derivatives in fibre bundles.
method Introducing a vertical endomorphism to construct covariant derivatives on vertical and horizontal distributions.
result Covariant derivatives can be constructed separately on vertical and horizontal distributions and then glued together.
Improved sampling for Diffusion Models by accounting for covariance.
problem Sampling quality degradation in few-step Diffusion Models.
method Covariance-aware sampler using Tweedie's formula and Fourier-space decomposition.
result Consistently superior samples compared to state-of-the-art samplers.
This paper improves computational efficiency in kernel ridge regression under covariate shift.
problem Covariate shift in nonparametric regression.
method Random projections in RKHS to reduce computational demands.
result Significant computational savings can be achieved without compromising learning performance under covariate shift.
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.
Study on Blaschke locus with covariance metric properties.
problem Riemannian structure of Blaschke locus.
method Analysis of Blaschke locus as a manifold, study of covariance metric properties.
result Geodesics in Blaschke locus have infinite length with respect to the covariance metric.
Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.
problem Understanding covariant derivatives of eigenfunctions on curved spaces.
method Analyzing the Laplace-Beltrami operator on Riemannian manifolds with constant curvature.
result Covariant derivatives of eigenfunctions are scalar multiples of the functions, and these scalars are polynomials in the eigenvalue.
Researchers create a family of conformally covariant operators.
problem Developing a comprehensive set of conformally covariant operators.
method Constructing a family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space.
result Symmetrization of ambient operators is formally self-adjoint.
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
The article defines conditions for a manifold to be conformal to an Einstein space.
problem Determining when a manifold is conformal to an Einstein space.
method Algorithmic conditions based on the metric tensor and the Weyl endomorphism.
result General necessary and sufficient conditions for a pseudo-Riemannian manifold to be conformal to an Einstein space.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorp…
New method needed for class prior estimation when covariates are reduced.
problem Class prior estimation fails under covariate shift when covariates are reduced.
method Propose a probing algorithm for class prior estimation.
result Provable transformations preserving covariate shift are necessary for class prior estimation.
We use the Nash embedding theorem to construct generators for the space of algebraic covariant derivative curvature tensors.
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
Paper explores geometry of covariance matrices using associated bundles.
problem Geometry of fixed-rank covariance matrices.
method Associated bundle approach to Bures--Wasserstein geometry.
result Established a one-to-one correspondence between geodesics.
This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
We consider the problem of joint estimation of structured inverse covariance matrices. We perform the estimation using groups of measurements with different covariances of the same unknown structure. Assuming the inverse covariances to span a low dimensional linear subspace in the space of symmetric matrices, our aim i…
Fast covariance calculation is required both for SLAM (e.g.~in order to solve data association) and for evaluating the information-theoretic term for different candidate actions in belief space planning (BSP). In this paper we make two primary contributions. First, we develop a novel general-purpose incremental covaria…
Optimally tackles covariate shift in RKHS-based nonparametric regression.
problem Covariate shift in nonparametric regression over RKHS.
method Two families of covariate shift problems defined using likelihood ratios. Minimax rate-optimal estimators for KRR and reweighted KRR.
result KRR is minimax rate-optimal and strictly sub-optimal compared to naive estimator under covariate shift.
In this contribution we describe an approach to evolve composite covariance functions for Gaussian processes using genetic programming. A critical aspect of Gaussian processes and similar kernel-based models such as SVM is, that the covariance function should be adapted to the modeled data. Frequently, the squared expo…
Paper estimates GMMs with unknown covariances using sparse regularization.
problem Estimating GMMs with unknown diagonal covariances from samples.
method Employed Beurling-LASSO (BLASSO) for sparse estimation of component means, covariances, and weights.
result Established non-asymptotic recovery guarantees with nearly parametric convergence rates.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space RqN, the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over RqN based on the R-matrix formalism, we…
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets…
The paper presents a novel approach to direct covariance function learning for Bayesian optimisation, with particular emphasis on experimental design problems where an existing corpus of condensed knowledge is present. The method presented borrows techniques from reproducing kernel Banach space theory (specifically m-k…
In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data. This decomposition imposes lower dimensional structure on the estimated covariance matrix, thus reducing the number of samples required for estimation. To allow a sm…
We study the problem of structured output learning from a regression perspective. We first provide a general formulation of the kernel dependency estimation (KDE) problem using operator-valued kernels. We show that some of the existing formulations of this problem are special cases of our framework. We then propose a c…
Efficiently estimates covariance for sparse functional data.
problem Sparse data in functional analysis.
method Random-knots and B-spline estimators for covariance function.
result Asymptotic pointwise covariance estimates for sparsified data.
Paper presents a new framework for covariance matrix estimation with geometric insights.
problem Challenges in covariance matrix estimation, especially in finding suitable models and efficient estimation methods.
method General framework for linear restrictions on different transformations of the covariance matrix, including matrix logarithm and its inverse.
result Yields an M-estimator with M-estimation allowing for straightforward asymptotic and finite sample analysis. Unified analysis of kernel-based methods under covariate shift.
problem Covariate shift in learning problems.
method Unified analysis of nonparametric methods in RKHS.
result Sharp convergence rates for general loss functions.
We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covaria…
Treeging combines regression trees and kriging for spatial and space-time prediction.
problem Improving spatial and space-time prediction accuracy.
method Combines regression trees with kriging's covariance structures.
result Treeging outperforms kriging and random forest in various scenarios.
New C-connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing C-connection, a Weyl connection that preserves conformal invariance. result Characterizes non-degenerate spaces conformal to Einstein spaces.
Novel covariance function improves Bayesian optimization efficiency.
problem Efficient global optimization of expensive black-box functions.
method Additive tree-structured covariance function and parallel optimization algorithm.
result Significantly outperforms state-of-the-art methods in conditional parameter optimization.
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.
Extends FJS analysis to general label spaces, including classification and regression.
problem Distribution shift in general label spaces, including covariate and label shifts.
method Proposes a framework for analyzing FJS in general label spaces and generalizes existing results.
result Generalizes FJS analysis to general label spaces, including classification and regression.
New method clusters high-dimensional data with anisotropic noise.
problem Clustering high-dimensional anisotropic mixtures with varying noise structures.
method Covariance Projected Spectral Clustering (COPO) method that projects data onto a low-dimensional space and reassigns clusters based on estimated covariances.
result COPO achieves minimax-optimal misclustering rates in Gaussian settings.