New estimator reduces kernel mean estimation error.
problem Kernel mean estimation in reproducing kernel Hilbert spaces.
method Corrupt data with known distributions and estimate kernel mean under the corrupted distribution.
result The marginalized kernel mean estimator achieves lower estimation error.
Adapts manifold structure for better clustering performance.
problem Lack of consideration for local manifold structure in existing multiple kernel k-means methods.
method Adopts manifold adaptive kernel to integrate local manifold structure of kernels.
result Proposed method outperforms state-of-the-art methods.
Kernel means are frequently used to represent probability distributions in machine learning problems. In particular, the well known kernel density estimator and the kernel mean embedding both have the form of a kernel mean. Unfortunately, kernel means are faced with scalability issues. A single point evaluation of the …
This note optimizes distributions using kernel mean embeddings with a new parameterization.
problem Optimizing distributions using kernel mean embeddings is challenging due to the difficulty of characterizing probability distribution vectors.
method Proposes a new parameterization of positive functions using kernel sums-of-squares to fit distributions in the MMD geometry.
result Distributions with kernel sum-of-squares densities are dense in the MMD geometry, allowing optimization in the finite-sample setting.
Paper improves clustering risk bounds for kernel k-means.
problem Improving clustering risk bounds for kernel k-means.
method Analyzes kernel k-means and Nyström approximation.
result Achieves nearly optimal excess clustering risk bound.
Kernel methods are studied in a mean field limit for high-dimensional data.
problem Analyzing kernel methods in high-dimensional data with many variables.
method Investigation of kernel methods in the mean field limit of interacting particle systems.
result Rigorous mean field limit of kernels and detailed analysis of the limiting reproducing kernel Hilbert space.
This paper provides a dictionary of closed-form kernel mean embeddings.
problem Challenges in deriving closed-form kernel mean embeddings.
method Comprehensive dictionary and practical tools for deriving new embeddings.
result Provides a Python library with minimal implementations of embeddings.
Bayesian deconditional embeddings solve complex function recovery.
problem Recovering original functions from conditional mean observations.
method Formalizes deconditional kernel mean embeddings as Bayesian inference, connects to task-transformed Gaussian processes.
result Establishes deconditional kernel means as posterior predictive mean, providing Bayesian interpretations and uncertainty.
Proposes a method to select diverse kernels for improved clustering performance.
problem Redundancy in selected kernels degrades clustering performance and efficiency.
method Selects diverse subset of kernels as representative kernels, optimizes combination coefficients using alternating minimization.
result Improves clustering performance and efficiency compared to existing methods.
A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an empirical average is the standard estimate for the true kernel mean. We show that this e…
Kernel methods are one of the mainstays of machine learning, but the problem of kernel learning remains challenging, with only a few heuristics and very little theory. This is of particular importance in methods based on estimation of kernel mean embeddings of probability measures. For characteristic kernels, which inc…
A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in…
Kernel K-means clusters probability distributions.
problem Clustering a sample of probability distributions.
method Mapping distributions to kernel mean embeddings in RKHS, then applying K-means.
result Effective unsupervised classification of probability distributions.
A new method learns hyperparameters for conditional kernel mean embeddings using Rademacher complexity bounds.
problem Hyperparameter tuning for conditional kernel mean embeddings is challenging and computationally expensive.
method Proposes a hyperparameter learning framework based on Rademacher complexity bounds for scalable kernel hyperparameter tuning.
result Demonstrates improved performance over competing methods and can incorporate deep neural network weights.
Kernel k-Means algorithm improves clustering of non-linear data.
problem Non-convexity of kernel k-Means objective function leads to local minima.
method Generalizes MM approach to solve non-convex problem in kernel and multi-kernel settings.
result Establishes strong consistency guarantees for Kernel Power k-Means.
Nyström method speeds up kernel k-means significantly.
problem Efficiency trade-off in kernel k-means.
method Nyström approach to reduce computational costs.
result Achieves same accuracy with fraction of computations.
Study on kernel methods in large-scale machine learning problems.
problem Large-scale machine learning with many interacting variables.
method Mean field limit analysis of kernels and their Hilbert spaces.
result Mean field convergence of empirical and infinite-sample solutions.
Kernel clustering algorithm improved for large datasets using incomplete Cholesky factorization.
problem Large memory usage in kernel-based clustering for large-scale datasets.
method Approximate the kernel matrix using incomplete Cholesky factorization and apply linear k-means clustering. result The proposed method achieves similar performance to kernel k-means clustering but handles large-scale datasets efficiently. Kernelized Taylor diagram visualizes data populations with fewer assumptions.
problem Limitations of Taylor diagram in capturing non-linear relationships and sensitivity to outliers.
method Proposes a kernelized version of the Taylor diagram that uses maximum mean discrepancy and kernel mean embedding.
result Kernelized Taylor diagram visualizes data populations with minimal assumptions of data distributions.
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.
New recursive algorithm estimates conditional kernel mean embeddings in Hilbert space.
problem Estimating conditional distributions in RKHS for supervised learning.
method Recursive algorithm in L2 space for conditional kernel mean map. result Strong L2 consistency of recursive estimator proved. Study guarantees convergence of mean shift mode estimation.
problem Ensuring reliable mode estimation in KDE using mean shift.
method Utilizes Łojasiewicz inequality to prove convergence rate.
result Extends convergence guarantees to biweight kernel.
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
Kernel k-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear k-means clustering algorithm. However, kernel k-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel …
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
Proposes a new K-means method for efficient clustering of nonlinear data.
problem Challenges of kernel K-means, including high memory usage and computational inefficiency.
method Combines linear and nonlinear approaches using explicit feature maps based on spectral analysis.
result Demonstrates Explicit Kernel Minkowski Weighted K-means (Explicit KMWK-means) reduces memory usage and improves efficiency.
Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.
problem Understanding convergence rates of maximum mean discrepancies for Farey sequences.
method Identifying positive-semidefinite kernels and their polynomial convergence rates.
result Polynomial convergence rate of maximum mean discrepancies of Farey sequences is equivalent to the Riemann hypothesis.
Algorithm learns interaction kernels for particle systems from data.
problem Understanding and modeling interactions in systems of interacting particles.
method Nonparametric algorithm using least squares with regularization, probabilistic error functional, and reproducing kernel Hilbert space convergence.
result The algorithm converges optimally and accurately learns interaction kernels.
Paper studies kernel hyperparameters for clustering, proposing an efficient search method.
problem Challenges in tuning kernel parameters for clustering, especially for RBF kernels.
method Derives a lower bound for RBF kernel parameters, proposes an efficient hyperparameter search algorithm.
result Proposes an efficient algorithm for hyperparameter search in kernel clustering, improving upon grid search.
IDK improves anomaly detection for points and groups without explicit learning.
problem Anomaly detection for points and groups using kernel methods.
method Isolation Distributional Kernel (IDK) addresses data independence and intractable dimensionality issues.
result IDK outperforms existing methods for both point and group anomaly detection.
Faster convergence of kernel mean embeddings using variance information.
problem Speeding up the convergence rate of kernel mean embeddings.
method Leveraging variance information in reproducing kernel Hilbert space and estimating variance from data.
result Efficiently estimate variance information from data to achieve distribution-agnostic convergence bounds.
Proposes a method to optimize class mean preservation in kernel-based feature spaces.
problem Optimizing the selection of kernel subspace for better performance.
method Component analysis method for kernel-based dimensionality reduction that optimally preserves class mean distances.
result Discriminant analysis version of the proposed method provides insights into feature space properties.
Kernel adaptive filters (KAF) are a class of powerful nonlinear filters developed in Reproducing Kernel Hilbert Space (RKHS). The Gaussian kernel is usually the default kernel in KAF algorithms, but selecting the proper kernel size (bandwidth) is still an open important issue especially for learning with small sample s…
Novel approach to OT using kernel mean embeddings controls overfitting and achieves dimension-free sample complexity.
problem Consistently estimate optimal transport plan from samples.
method Pose OT as learning kernel mean embedding, employ MMD regularization.
result ε-optimal recovery of transport plan and map with dimension-free sample complexity.
We introduce two kernels that extend the mean map, which embeds probability measures in Hilbert spaces. The generative mean map kernel (GMMK) is a smooth similarity measure between probabilistic models. The latent mean map kernel (LMMK) generalizes the non-iid formulation of Hilbert space embeddings of empirical distri…
Efficiently approximates kernel mean embeddings using Nyström method.
problem Computational cost of kernel mean embeddings in large-scale settings.
method Nyström method for approximating a small random subset of the dataset.
result Upper bound on approximation error with sufficient subsample size conditions.
The paper examines the optimality of kernel methods in high-dimensional clustering.
problem Understanding the optimality of kernel methods in high-dimensional data clustering.
method High-dimensional Gaussian clustering, exponential kernel function, kernel k-means, semi-definite relaxation.
result The exponential kernel function optimally recovers clusters in high-dimensional data, matching information-theoretic limits up to a factor of √2.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
New algorithm detects changes in Markov kernels with unknown post-change kernel.
problem Detecting changes in Markov kernels with unknown post-change kernel.
method Developed a new change detection algorithm assuming uniform ergodicity.
result Derived upper and lower bounds on mean delay and time between false alarms.
A new method optimizes kernels for GANs and SVMs using mean-field theory.
problem Optimizing kernels for GANs and SVMs in a distributionally robust setting.
method Distributionally robust optimization, Monte-Carlo SAA, particle SGD, mean-field analysis.
result The method improves kernel learning for hypothesis testing and achieves better test power.
A new Mean Shift variant converges after a finite number of iterations for specific kernel shapes.
problem Improving Mean Shift algorithm convergence for specific kernel shapes.
method Developed a novel Mean Shift variant with a triangular kernel profile.
result The new Mean Shift variant converges after a finite number of iterations.
Kernel-based K-means clustering has gained popularity due to its simplicity and the power of its implicit non-linear representation of the data. A dominant concern is the memory requirement since memory scales as the square of the number of data points. We provide a new analysis of a class of approximate kernel methods…
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.
SimpleMKKM improves multi-kernel clustering efficiency.
problem Efficient multi-kernel clustering.
method Re-formulated minimization-maximization problem into a smooth minimization, solved with gradient descent.
result Outperforms state-of-the-art multi-kernel clustering alternatives.
The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rel…
Paper introduces RKHM and KME for richer data analysis.
problem Lack of rich data structures in kernel methods.
method Proposes RKHM and KME for functional data analysis.
result RKHM captures structural properties in functional data.
We connect shift-invariant characteristic kernels to infinitely divisible distributions on Rd. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, usi…
The paper studies kernel smoothing and mean shift for directional data, deriving convergence rates and mode estimation.
problem Statistical and computational problems of kernel smoothing for directional data.
method Generalization of mean shift to directional data, derivation of convergence rates, and investigation of mode estimation.
result Statistical convergence rates of directional KDE and its derivatives, ascending property of directional mean shift, and mode estimation.