Robust kernel CCA method detects outliers and improves performance.
problem Kernel CO and CCO sensitivity to contaminated data.
method Proposed robust kernel CO and CCO, derived IF for CCA, robust kernel CCA method.
result Robust kernel CCA method performs better than standard kernel CCA for ideal and contaminated data.
To the best of our knowledge, there are no general well-founded robust methods for statistical unsupervised learning. Most of the unsupervised methods explicitly or implicitly depend on the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). They are sensitive to contaminated data, …
Active data collection improves convergence rates in operator learning.
problem Improving convergence rates in operator learning with linear target and stochastic input.
method Active data collection strategies with mean-zero stochastic process and continuous covariance kernels.
result Achieves arbitrarily fast error convergence rates with eigenvalue decay of covariance kernels.
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…
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.
Kernel operators help detect patterns in complex data.
problem Detecting long-lived coherent patterns in high-dimensional time-series data.
method Dominant eigenfunctions of kernel transfer operators combined with gradient-based optimization.
result Effective detection of long-lived coherent patterns in high-dimensional time-series data.
Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.
problem Developing theory for Schrödinger semigroups on Riemannian manifolds.
method Sobolev spaces, heat kernels, differential operators, Wiener measure, Dynkin and Kato potentials.
result Properties and continuity of covariant Schrödinger semigroups established.
This paper provides a functional analytic foundation for singular value decomposition of RKHS operators.
problem Singular value decomposition of operators on RKHSs.
method Functional analytic approach, extending matrix eigenvalue problems to RKHS operators.
result Solid foundation and extension of singular value decomposition to RKHS operators.
NTKs explain GNNs' alignment for graph prediction.
problem Understanding GNNs' alignment for graph prediction.
method Analyzing NTKs and alignment in GNNs, focusing on cross-covariance.
result Optimizing alignment in GNNs optimizes graph representation.
Method selects features via kernel-based independence measures.
problem Feature selection in high-dimensional data.
method Optimization of conditional covariance trace.
result Method outperforms other feature selection algorithms.
This paper uses random Fourier features to simplify latent force models and convolved Gaussian processes.
problem Expensive covariance matrix calculation in latent force models due to double integrals.
method Approximates double integrals using random Fourier features to obtain simpler analytical expressions.
result Simplified analytical expressions for covariance functions, leading to faster computation.
We analyze the size of the dictionary constructed from online kernel sparsification, using a novel formula that expresses the expected determinant of the kernel Gram matrix in terms of the eigenvalues of the covariance operator. Using this formula, we are able to connect the cardinality of the dictionary with the eigen…
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.
RKUM is an R package for robust kernel-based unsupervised methods.
problem Robust analysis under contaminated or noisy data conditions.
method Robust kernel covariance and cross-covariance operators using generalized loss functions.
result RKUM reduces sensitivity to contamination and effectively identifies outliers.
A new kernel-based nonconformity score improves multivariate prediction regions.
problem Tackling the challenge of compressing multivariate residual vectors into scalars while preserving geometric structure.
method Introducing a Multivariate Kernel Score (MKS) that decomposes into an anisotropic MMD, providing finite-sample coverage guarantees and convergence rates.
result The MKS produces prediction regions that explicitly adapt to geometric structure, reducing volume compared to ellipsoidal baselines.
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
New method tests fit between source and target populations.
problem Testing goodness-of-fit under covariate shift.
method Truncated importance-weighting kernel ridge regression with multiplier bootstrap.
result Valid and sharp confidence sets for regression function.
Let G be a finite connected simple graph. We define the moduli space of conformal structures on G. We propose a definition of conformally covariant operators on graphs, motivated by [25]. We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the …
A new distance metric compares probability distributions using kernel covariance operators.
problem Comparing probability distributions in machine learning tasks.
method Introduces a novel distance metric based on Schatten norm of kernel covariance operators.
result The new distance metric is more discriminative and robust to hyperparameters.
Extends Mahalanobis distance to Banach spaces for anomaly detection.
problem Anomaly detection in infinite-dimensional spaces.
method Generalizes Mahalanobis distance to Banach spaces via Cameron-Martin norm and variance norm.
result Kernelized nearest-neighbour Mahalanobis distance outperforms traditional methods for time series novelty detection.
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.
In this paper we present a nonparametric method for extending functional regression methodology to the situation where more than one functional covariate is used to predict a functional response. Borrowing the idea from Kadri et al. (2010a), the method, which support mixed discrete and continuous explanatory variables,…
New asymmetric kernel methods improve feature learning.
problem Improving feature learning with asymmetric kernels.
method Coupled covariance eigenproblem and Nyström method.
result Empirical evaluations show benefits of KSVD.
Develops efficient inference for noise heterogeneity in machine learning models.
problem Downstream procedures based on residuals can be biased in additive noise models.
method Semiparametrically efficient inference using a novel Hilbert-valued one-step estimator.
result Constructs tests and confidence intervals for residual independence and goodness of fit.
Imaging genetic research has essentially focused on discovering unique and co-association effects, but typically ignoring to identify outliers or atypical objects in genetic as well as non-genetics variables. Identifying significant outliers is an essential and challenging issue for imaging genetics and multiple source…
Paper studies regularized KKL divergence for distributions with disjoint supports.
problem Inability of original KKL divergence to handle distributions with disjoint supports.
method Proposes a regularized variant of KKL divergence, derives bounds, and provides closed-form expression.
result Regularized KKL divergence is well-defined for all distributions and has finite-sample bounds.
A streaming algorithm estimates quadratic covariation from financial data efficiently.
problem Estimating quadratic covariation from ultra-high-frequency financial data with limited memory.
method Formulated multi-scale, realized kernel, pre-averaging, and modulated realized covariance estimators with fixed bandwidth.
result Fixed bandwidth estimators require higher bandwidth for positive semidefiniteness.
The paper proves conditions for the triviality of L2-harmonic forms on Riemannian manifolds.
problem Conditions for the triviality of L2-harmonic forms on Riemannian manifolds. method Study of a covariant Schrödinger operator HX,V and its L2-kernel. result Sufficient conditions for the triviality of the L2-kernel of HX,V. Proposes a new divergence measure for probability distributions.
problem Challenges in estimating divergences from empirical samples.
method Embeds data into RKHS, computes Jensen-Shannon divergence between covariance operators.
result Establishes RJSD as a lower bound on Jensen-Shannon divergence, enabling variational estimation.
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.
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…
Kernel ridge regression for causal inference with missing data.
problem Estimating treatment effects with missing data in selected samples.
method Kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects.
result Uniform consistency and finite sample rates for continuous treatment, root-n consistency for discrete treatment.
We study compressing empirical measures in finite RKHSs using convex optimization.
problem Efficiently approximating empirical measures in high-dimensional spaces.
method Convex optimization and lower bounds on ball size.
result High probability lower bounds on ball size under various conditions.
The study examines the geometry of Lichnerowicz Laplacian's kernel on various spaces.
problem Understanding the kernel of the Lichnerowicz Laplacian on different types of spaces.
method Analytical method of Bochner to prove vanishing theorems for null space of Laplace operator.
result Applications to theories of infinitesimal Einstein deformations and stability of Einstein manifolds.
Study reveals KLMS algorithms as simplified GP regression models.
problem Understanding KLMS algorithms and their performance differences.
method Examined the relationship between online Gaussian process regression and KLMS algorithms.
result KLMS algorithms correspond to specific cases of a parametric model of posterior covariance.
Convolution and pooling improve kernel methods in image classification.
problem Understanding the interplay between approximation and generalization in convolutional architectures.
method Characterized RKHS of kernels with convolution, pooling, and downsampling, computed generalization error.
result Convolution and pooling operations trade off approximation with generalization power.
Study spectral properties of graph Laplacian for manifold data.
problem Understanding spectral properties of graph Laplacian for manifold data.
method Non-asymptotic error bounds on spectral properties of empirical graph Laplacian.
result Eigenvalues and eigenspaces of empirical graph Laplacian are close to Laplace-Beltrami operator of manifold.
We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism o…
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension n≥3, there exist many metr…
New method tests conditional independence using spectral representations.
problem Untestable conditional independence in many settings.
method Spectral representations of partial covariance operators, bi-level contrastive learning.
result Asymptotic validity and power guarantees for CI testing.
Paper provides unbiased spectral moment estimates from finite data.
problem Challenges in estimating spectral moments from limited data.
method Dynamic programming approach to estimate spectral moments of kernel integral operator.
result Demonstrates consistency with theoretical spectra and practical utility in neural networks.
A new data-adaptive prior stabilizes kernel learning in operators.
problem Learning kernels in operators from data is ill-posed due to nonlocal dependence.
method Introduces a data-adaptive prior to stabilize the Bayesian posterior mean.
result The data-adaptive prior achieves a stable posterior with small noise limits.
Entropy analysis via kernel methods for probabilistic inference.
problem Entropy analysis of probability distributions.
method Kernel methods and reproducing kernel Hilbert spaces for entropy estimation.
result New upper-bounds on log partition functions for probabilistic inference.
Investigates kernel regression rates without assuming polynomial eigenvalue decay.
problem Achieving minimax rates without strict assumptions on kernel eigenvalue decay.
method Examines kernel regularization methods under weak assumptions on eigenvalue decay.
result Achieves minimax convergence rates under less restrictive conditions.
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.
Using Weitzenböck techniques on any compact Riemannian spin manifold we derive a general inequality depending on a real parameter and joining the spectrum of the Dirac operator with terms depending on the Ricci tensor and its first covariant derivatives. The discussion of this inequality yields vanishing theorems for t…
This work extends Gaussian process priors to neural operators for function space mappings.
problem Improving uncertainty quantification in deep neural networks.
method Extending Gaussian process priors to neural operators with conditions for convergence and computation of covariance functions.
result Arbitrary-depth neural operators with Gaussian kernels converge to function-valued GPs, enabling posterior computation in regression scenarios.
In genome-wide interaction studies, to detect gene-gene interactions, most methods are divided into two folds: single nucleotide polymorphisms (SNP) based and gene-based methods. Basically, the methods based on the gene are more effective than the methods based on a single SNP. Recent years, while the kernel canonical …