Quantum kernels can be efficiently embedded into classical feature spaces.
problem Can all quantum kernels be efficiently embedded into classical feature spaces?
method Invoking computational universality and using techniques like random Fourier features, the authors show that certain classes of quantum kernels can be efficiently embedded.
result For shift-invariant and composition kernels, embedding quantum kernels are universal and efficient.
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.
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.
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…
Kernel embeddings help estimate causal effects from observational data.
problem Estimating causal effects from observational data with confounding variables.
method Kernel embeddings in reproducing kernel Hilbert spaces (RKHS).
result Robust nonparametric framework for causal inference.
New embeddings for manifolds using heat kernels.
problem Constructing canonical conformal embeddings for manifolds.
method Employing heat kernel embedding from Bérard-Besson-Gallot'94 to find canonical conformal embeddings.
result Intrinsic construction of canonical conformal embeddings with dimensions growing exponentially with t. A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the re…
A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
Kernel embeddings separate distinct probability distributions, simplifying testing.
problem Testing equality of non-atomic probability distributions.
method Kernel covariance embeddings and Gaussian measures in reproducing kernel Hilbert spaces.
result Testing for singularity between Gaussian measures is equivalent to testing for equality of non-atomic probability distributions.
New graph kernel scales well with graph size and number, achieving state-of-the-art performance.
problem Graph kernels lose structure information when representing graphs.
method Proposes a positive-definite global alignment graph kernel using random features and random graph embeddings.
result Achieves quasi-linear scalability with respect to graph size and number.
Kernel discriminant analysis uses nonlinear embeddings to improve classification.
problem Limited effectiveness of linear discriminant analysis in capturing nonlinear features.
method Study of nonlinear embeddings in kernel discriminant analysis using polynomial and Gaussian kernels, solving generalized eigenvalue problems.
result Polynomial and Gaussian discriminants capture class differences through population moments and randomized projections.
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.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
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.
In recent years, there have been significant efforts on mitigating unethical demographic biases in machine learning methods. However, very little is done for kernel methods. In this paper, we propose a new fair kernel regression method via fair feature embedding (FKR-F2E) in kernel space. Motivated by prior works on…
Conditional kernel mean embeddings form an attractive nonparametric framework for representing conditional means of functions, describing the observation processes for many complex models. However, the recovery of the original underlying function of interest whose conditional mean was observed is a challenging inferenc…
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.
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.
Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures μ from some set M to functions in a reproducing kernel Hilbert space (RKHS) with kernel k. The RKHS distance of two mapped measures is a semi-metric dk over M. We study three questions. (I) For a…
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.
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. Gaussian kernel fails on circle and related spaces.
problem Gaussian kernel's positive definiteness on non-Euclidean spaces.
method Analyzing the Gaussian kernel on the circle and related metric spaces.
result Gaussian kernel is not positive definite on the circle or spaces admitting circle embeddings.
New test for conditional independence using kernel embeddings.
problem Testing conditional independence in high-dimensional settings.
method Analytic kernel embeddings, asymptotic distribution.
result New test outperforms existing methods in high-dimensional settings.
We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based …
Paper proposes a new method to learn distribution kernels via entropy maximization.
problem Challenges in applying kernel methods to distribution regression tasks.
method Proposes a novel objective for unsupervised learning of data-dependent distribution kernels based on entropy maximization.
result Demonstrates the effectiveness of the learned kernel across different modalities.
Paper generalizes kernel mean embedding to von Neumann-algebra-valued measures.
problem Analyzing complex multivariate distributions and quantum mechanics.
method Generalizes kernel mean embedding to von Neumann-algebra-valued measures in reproducing kernel Hilbert modules.
result Injectivity and universality of the generalized KME are confirmed.
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.
A novel kernel-based test detects equality versus singularity of two probability measures.
problem Detecting equality versus singularity of two probability distributions.
method Combines kernel mean and kernel covariance embeddings to construct a likelihood ratio test statistic.
result The test statistic satisfies a '0/\infty' law, vanishing under the null and diverging under the alternative.
For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded imag…
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 …
Maps embed manifolds using heat kernels of connection Laplacian.
problem Embedding manifolds in Euclidean space.
method Using heat kernels of the connection Laplacian and truncated heat kernels.
result Maps can be made arbitrarily close to isometries.
In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space Rd. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…
Conditional kernel mean embeddings are nonparametric models that encode conditional expectations in a reproducing kernel Hilbert space. While they provide a flexible and powerful framework for probabilistic inference, their performance is highly dependent on the choice of kernel and regularization hyperparameters. Neve…
Improved learning theory for kernel distribution regression with two-stage sampling.
problem Distribution regression problem and two-stage sampling setting.
method Kernel methods, near-unbiased condition, new error bounds, convergence rates.
result Strictly improved convergence rates for three important classes of kernels.
A new method embeds data using Gaussian processes based on the heat kernel.
problem Embedding high-dimensional data in a low-dimensional space.
method Computing embeddings based on the Karhunen-Loève expansion of the heat kernel.
result The embedding approximates diffusion distances and is robust to outliers.
The kernel embedding algorithm is an important component for adapting kernel methods to large datasets. Since the algorithm consumes a major computation cost in the testing phase, we propose a novel teacher-learner framework of learning computation-efficient kernel embeddings from specific data. In the framework, the h…
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.
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be ex…
We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set opt…
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…
Kernel method embeds noisy datasets, capturing shared structures.
problem Limited power in capturing nonlinear structures, noisiness, high-dimensionality, and interpretability issues.
method Kernel spectral joint embeddings using duo-landmark integral operators.
result Consistent recovery of low-dimensional noiseless signals and convergence to eigenfunctions of integral operators.
A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the spac…
Estimates class prior for unlabeled data using kernel embedding.
problem Estimating class prior in PU learning scenario where only positive and full population samples are available.
method Direct estimator based on distribution matching and kernel embedding in Reproducing Kernel Hilbert Space.
result Asymptotic consistency and explicit deviation bound for the estimator.
Alternative proof of coisotropic embedding theorem for pre-symplectic manifolds.
problem Proving the coisotropic embedding theorem for pre-symplectic manifolds.
method Recast geometric choice of connection as algebraic embedding into cotangent bundle, identify symplectic thickening as submanifold of Hamiltonian momenta conjugate to kernel directions.
result Alternative proof of the coisotropic embedding theorem.
Kernel fusion is a popular and effective approach for combining multiple features that characterize different aspects of data. Traditional approaches for Multiple Kernel Learning (MKL) attempt to learn the parameters for combining the kernels through sophisticated optimization procedures. In this paper, we propose an a…
Proposes estimators for complex dose-response curves using kernel methods.
problem Estimating complex dose-response curves with continuous treatments, mediators, and covariates.
method Kernel ridge regression with sequential kernel embedding technique.
result Simple estimators for mediated and time-varying dose response curves with nonasymptotic uniform rates.
Kernel-spectral embedding learns low-dim. structures from noisy data.
problem Learning low-dimensional nonlinear structures from high-dimensional noisy data.
method Adaptive bandwidth spectral embedding using integral operators.
result Convergence to noiseless embeddings and eigenfunctions of integral operators.
Develops a rigorous theory for conditional mean embeddings.
problem Efficient conditioning of probability distributions in RKHSs.
method Mathematical theory for both centred and uncentred covariance operators.
result Significantly weakens conditions for applicability of CMEs.