Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class and within-class covariance operators. The polynomial discriminants are shown to capture the class difference through the population moments explicitly. For approximation of the Gaussian discriminant, we use a particular representation of the Gaussian kernel by utilizing the exponential generating function for Hermite polynomials. We also show that the Gaussian discriminant can be approximated using randomized projections of the data. Our results illuminate how the data distribution and the kernel interact in determination of the nonlinear embedding for discrimination, and provide a guideline for choice of the kernel and its parameters.
Kernel approximation methods create explicit, low-dimensional kernel feature maps to deal with the high computational and memory complexity of standard techniques. This work studies a supervised kernel learning methodology to optimize such mappings. We utilize the Discriminant Information criterion, a measure of class separability with a strong connection to Discriminant Analysis. By generalizing this measure to cover a wider range of kernel maps and learning settings, we develop scalable methods to learn kernel features with high discriminant power. Experimental results on several datasets sh…
In many artificial intelligence and computer vision systems, the same object can be observed at distinct viewpoints or by diverse sensors, which raises the challenges for recognizing objects from different, even heterogeneous views. Multi-view discriminant analysis (MvDA) is an effective multi-view subspace learning method, which finds a discriminant common subspace by jointly learning multiple view-specific linear projections for object recognition from multiple views, in a non-pairwise way. In this paper, we propose the kernel version of multi-view discriminant analysis, called kernel multi-…
Kernel alignment measures the degree of similarity between two kernels. In this paper, inspired from kernel alignment, we propose a new Linear Discriminant Analysis (LDA) formulation, kernel alignment LDA (kaLDA). We first define two kernels, data kernel and class indicator kernel. The problem is to find a subspace to maximize the alignment between subspace-transformed data kernel and class indicator kernel. Surprisingly, the kernel alignment induced kaLDA objective function is very similar to classical LDA and can be expressed using between-class and total scatter matrices. This can be extend…
Eskişehir Teknik Üniversitesi（埃斯基谢希尔技术大学）；Erasmus University Rotterdam（鹿特丹伊拉斯姆斯大学）
Kernels are often developed and used as implicit mapping functions that show impressive predictive power due to their high-dimensional feature space representations. In this study, we gradually construct a series of simple feature maps that lead to a collection of interpretable low-dimensional kernels. At each step, we keep the original features and make sure that the increase in the dimension of input data is extremely low, so that the resulting discriminant functions remain interpretable and amenable to fast training. Despite our persistence on interpretability, we obtain high accuracy resul…
Department of Engineering, ECE, Aarhus University（奥胡斯大学工程系（ECE））
The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on the kernel matrix defines an explicit nonlinear mapping of the input data representations to a subspace of the kernel space, which can be used for directly applying linear methods. However, the selection of the kernel subspace is crucial for the performance of the proceeding processing steps. In this paper, we propose a component analysis method for kernel-bas…
We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparam…
This is a detailed tutorial paper which explains the Fisher discriminant Analysis (FDA) and kernel FDA. We start with projection and reconstruction. Then, one- and multi-dimensional FDA subspaces are covered. Scatters in two- and then multi-classes are explained in FDA. Then, we discuss on the rank of the scatters and the dimensionality of the subspace. A real-life example is also provided for interpreting FDA. Then, possible singularity of the scatter is discussed to introduce robust FDA. PCA and FDA directions are also compared. We also prove that FDA and linear discriminant analysis are equ…
We present a new method which generalizes subspace learning based on eigenvalue and generalized eigenvalue problems. This method, Roweis Discriminant Analysis (RDA), is named after Sam Roweis to whom the field of subspace learning owes significantly. RDA is a family of infinite number of algorithms where Principal Component Analysis (PCA), Supervised PCA (SPCA), and Fisher Discriminant Analysis (FDA) are special cases. One of the extreme special cases, which we name Double Supervised Discriminant Analysis (DSDA), uses the labels twice; it is novel and has not appeared elsewhere. We propose a d…
Phonexia（Phonexia公司）；Brno University of Technology（布尔诺理工大学）；Omilia - Conversational Intelligence（Omilia对话智能公司）
In speaker recognition, where speech segments are mapped to embeddings on the unit hypersphere, two scoring backends are commonly used, namely cosine scoring or PLDA. Both have advantages and disadvantages, depending on the context. Cosine scoring follows naturally from the spherical geometry, but for PLDA the blessing is mixed -- length normalization Gaussianizes the between-speaker distribution, but violates the assumption of a speaker-independent within-speaker distribution. We propose PSDA, an analogue to PLDA that uses Von Mises-Fisher distributions on the hypersphere for both within and …
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-F$^2$E) in kernel space. Motivated by prior works on feature selection in kernel space and feature processing for fair machine learning, we propose to learn fair feature embedding functions that minimize demographic discrepancy of feature distributions in kernel space. Compared to the state-of-the-art fair kernel regression method…
INRIA（法国国家信息与自动化研究所）；PSL Research University（PSL研究大学）
Kernel mean embeddings are a popular tool that consists in representing probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. When the kernel is characteristic, mean embeddings can be used to define a distance between probability measures, known as the maximum mean discrepancy (MMD). A well-known advantage of mean embeddings and MMD is their low computational cost and low sample complexity. However, kernel mean embeddings have had limited applications to problems that consist in optimizing distributions, due to the difficulty of characterizin…
IBM Research（IBM研究院）；CMU（卡内基梅隆大学）；WUSTL（华盛顿大学圣路易斯分校）；GMU（乔治梅森大学）；UDEL（特拉华大学）；Huawei（华为）
Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the alignment of geometric node embeddings of graphs, yield state-of-the-art performance. However, these graph kernels are not necessarily positive-definite. More importantly, computing the graph kernel matrix will have at least quadratic {time} complexity in terms of the number and the s…
University of Sydney（悉尼大学）；CSIRO（澳大利亚联邦科学与工业研究组织）；NVIDIA（英伟达）
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 inference task. We formalize deconditional kernel mean embeddings as a solution to this inverse problem, and show that it can be naturally viewed as a nonparametric Bayes' rule. Critically, we introduce the notion of task transformed Gaussian processes and establish deconditional kernel …
Institute of Analytics and Data Science, Kennesaw State University（肯尼索州立大学分析与数据科学研究所）；Department of Information Technology, Kennesaw State University（肯尼索州立大学信息技术系）
In this paper, we propose a novel supervised learning method that is called Deep Embedding Kernel (DEK). DEK combines the advantages of deep learning and kernel methods in a unified framework. More specifically, DEK is a learnable kernel represented by a newly designed deep architecture. Compared with pre-defined kernels, this kernel can be explicitly trained to map data to an optimized high-level feature space where data may have favorable features toward the application. Compared with typical deep learning using SoftMax or logistic regression as the top layer, DEK is expected to be more gene…
We present an operator-free, measure-theoretic approach to the conditional mean embedding (CME) as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of unconditional distributions has been defined rigorously, the existing operator-based approach of the conditional version depends on stringent assumptions that hinder its analysis. We overcome this limitation via a measure-theoretic treatment of CMEs. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough theoretical analysis thereof, including univ…
University of Sydney（悉尼大学）；Data61, CSIRO（联邦科学与工业研究组织数据61研究所）；Australian National University（澳大利亚国立大学）
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. Nevertheless, current hyperparameter tuning methods predominantly rely on expensive cross validation or heuristics that is not optimized for the inference task. For conditional kernel mean embeddings with categorical targets and arbitrary inputs, we propose a hyperparameter learning …
School of Informatics（信息学院）；University of Edinburgh（爱丁堡大学）；FU Berlin（柏林自由大学）
We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a model free, non-parametric approach for calculation of an approximate solution to the control problem. This formulation admits a decomposition of the problem into an invariant and task dependent component. Consequently, we make much more efficient use of the sample data compared to previous sample based approaches in this domain, e.g., by allowing sample re-use across tasks. Numerical ex…
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）；Microsoft IDC（微软IDC）
The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing statistical OT as that of learning the transport plan's kernel mean embedding from sample based estimates of marginal embeddings. The proposed estimator controls overfitting by employing maximum mean discrepancy based regularization, which is complementary to $ϕ$-divergence (entropy) based regularization popularly employed in existing estimators. A key result is t…
CREST（法国 Crest 研究中心）；ENSAE（法国国家统计与经济管理学院）；Facebook AI Research（脸书人工智能研究院）；Université Paris-Dauphine（巴黎多芬大学）；Technion（以色列理工学院）
We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of locations. We obtain its asymptotic distribution under the null hypothesis of conditional independence and design a consistent statistical test from it. We conduct a series of experiments showing that our new test outperforms state-of-the-art methods both in terms of type-I and type-II errors even in the high dimensional setting.
We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not…
Mahidol University（玛希隆大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Institute of Statistical Mathematics（统计数学研究所）；Pennsylvania State University（宾夕法尼亚州立大学）
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 extended to probability measures. It can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kern…
Università di Genova（热那亚大学）；École normale supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；MIT（麻省理工学院）；IIT（意大利理工学院）
Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss ap…
Current meta-learning approaches focus on learning functional representations of relationships between variables, i.e. on estimating conditional expectations in regression. In many applications, however, we are faced with conditional distributions which cannot be meaningfully summarized using expectation only (due to e.g. multimodality). Hence, we consider the problem of conditional density estimation in the meta-learning setting. We introduce a novel technique for meta-learning which combines neural representation and noise-contrastive estimation with the established literature of conditional…
This is a tutorial and survey paper for Locally Linear Embedding (LLE) and its variants. The idea of LLE is fitting the local structure of manifold in the embedding space. In this paper, we first cover LLE, kernel LLE, inverse LLE, and feature fusion with LLE. Then, we cover out-of-sample embedding using linear reconstruction, eigenfunctions, and kernel mapping. Incremental LLE is explained for embedding streaming data. Landmark LLE methods using the Nystrom approximation and locally linear landmarks are explained for big data embedding. We introduce the methods for parameter selection of numb…
The University of Sydney（悉尼大学）；CSIRO（澳大利亚联邦科学与工业研究组织）；NVIDIA（英伟达）
In likelihood-free settings where likelihood evaluations are intractable, approximate Bayesian computation (ABC) addresses the formidable inference task to discover plausible parameters of simulation programs that explain the observations. However, they demand large quantities of simulation calls. Critically, hyperparameters that determine measures of simulation discrepancy crucially balance inference accuracy and sample efficiency, yet are difficult to tune. In this paper, we present kernel embedding likelihood-free inference (KELFI), a holistic framework that automatically learns model hyper…
SZTAKI: Institute for Computer Science and Control（SZTAKI：计算机科学与控制研究所）；Eötvös Loránd Research Network（厄特沃什·罗兰研究网络）
In this paper we suggest two statistical hypothesis tests for the regression function of binary classification based on conditional kernel mean embeddings. The regression function is a fundamental object in classification as it determines both the Bayes optimal classifier and the misclassification probabilities. A resampling based framework is presented and combined with consistent point estimators of the conditional kernel mean map, in order to construct distribution-free hypothesis tests. These tests are introduced in a flexible manner allowing us to control the exact probability of type I e…
RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所高级智能项目中心（AIP））；Guangdong University of Technology（广东工业大学）
Although the convolutional neural networks (CNNs) have become popular for various image processing and computer vision task recently, it remains a challenging problem to reduce the storage cost of the parameters for resource-limited platforms. In the previous studies, tensor decomposition (TD) has achieved promising compression performance by embedding the kernel of a convolutional layer into a low-rank subspace. However the employment of TD is naively on the kernel or its specified variants. Unlike the conventional approaches, this paper shows that the kernel can be embedded into more general…
Google Brain（谷歌大脑）；Georgia Institute of Technology（佐治亚理工学院）；University College London（伦敦大学学院）；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We establish a connection between kernel exponential family estimation and M…
Carnegie Mellon University（卡内基梅隆大学）；PathAI；Lawrence Livermore National Laboratories（劳伦斯利弗莫尔国家实验室）
Gaussian Processes (GPs) are known to provide accurate predictions and uncertainty estimates even with small amounts of labeled data by capturing similarity between data points through their kernel function. However traditional GP kernels are not very effective at capturing similarity between high dimensional data points. Neural networks can be used to learn good representations that encode intricate structures in high dimensional data, and can be used as inputs to the GP kernel. However the huge data requirement of neural networks makes this approach ineffective in small data settings. To sol…
The Chinese University of Hong Kong（香港中文大学）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）
Although nonstationary data are more common in the real world, most existing causal discovery methods do not take nonstationarity into consideration. In this letter, we propose a kernel embedding-based approach, ENCI, for nonstationary causal model inference where data are collected from multiple domains with varying distributions. In ENCI, we transform the complicated relation of a cause-effect pair into a linear model of variables of which observations correspond to the kernel embeddings of the cause-and-effect distributions in different domains. In this way, we are able to estimate the caus…
In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the solution of an infinite dimensional analogue of a semi-definite program. This embedding is adaptive and non-linear. We discuss this problem both with weak and strong smoothness assumptions about the learned kernel. A main feature of our approach is the existence of an out-of-sample …
Freie Universität Berlin（柏林自由大学）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料与能源研究中心）；Universiteit Leiden（莱顿大学）；LIACS, Universiteit Leiden（莱顿大学高级计算机科学研究所）
Graduate University of Advanced Studies (SOKENDAI)（高等研究生院（SOKENDAI））；The Institute of Statistical Mathematics（统计数学研究所）；The University of Adelaide（阿德莱德大学）
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 include most commonly used ones, the kernel mean embedding uniquely determines its probability measure, so it can be used to design a powerful statistical testing framework, which includes nonparametric two-sample and independence tests. In practice, however, the performance of thes…
We propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling exponentially many isomorphic forms. Bhattacharyya kernel constructed between these functionals significantly outperforms the state-of-the-art graph kernels on 3 out of the 4 standard benchmark graph classification datasets, demonstrating the superiority of our approach. The propo…
The fundamental goal of self-supervised learning (SSL) is to produce useful representations of data without access to any labels for classifying the data. Modern methods in SSL, which form representations based on known or constructed relationships between samples, have been particularly effective at this task. Here, we aim to extend this framework to incorporate algorithms based on kernel methods where embeddings are constructed by linear maps acting on the feature space of a kernel. In this kernel regime, we derive methods to find the optimal form of the output representations for contrastiv…
A Dictionary of Closed-Form Kernel Mean Embeddings
University College London（伦敦大学学院）；University of Tübingen（蒂宾根大学）；Lappeenranta–Lahti University of Technology LUT（拉彭兰塔-拉赫蒂理工大学（LUT））；Yahoo Research（雅虎研究院）
NTT Network Service Systems Laboratories, NTT Corporation（NTT网络服务系统实验室，NTT公司）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Center for Data Science, Ehime University（爱媛大学数据科学中心）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）；Faculty of Science and Technology, Keio University（庆应义塾大学科学与技术学部）
Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and extract structural properties in such as functional data. We show a branch of theories for RKHM to apply to data analysis, including the representer…
CentraleSupélec（中央理工-高等电力学院）；Inria Saclay（法国国家信息与自动化研究所萨克雷中心）；University of Paris-Saclay（巴黎-萨克雷大学）
Learning representations of nodes in a low dimensional space is a crucial task with many interesting applications in network analysis, including link prediction and node classification. Two popular approaches for this problem include matrix factorization and random walk-based models. In this paper, we aim to bring together the best of both worlds, towards learning latent node representations. In particular, we propose a weighted matrix factorization model which encodes random walk-based information about the nodes of the graph. The main benefit of this formulation is that it allows to utilize …
Recent results on optimization and generalization properties of neural networks showed that in a simple two-layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines the convergence of the optimization during training. Such analyses also provide upper bounds on the generalization error. We experimentally investigate the implications of these results to deeper networks via embeddings. We regard the layers preceding the final hidden layer as producing different representations of the input data which are then fed to the two-layer model. We show th…
Australian National University（澳大利亚国立大学）；NICTA（澳大利亚国家信息与通信技术研究院）
Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g, support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as …
We lay theoretical foundations for new database release mechanisms that allow third-parties to construct consistent estimators of population statistics, while ensuring that the privacy of each individual contributing to the database is protected. The proposed framework rests on two main ideas. First, releasing (an estimate of) the kernel mean embedding of the data generating random variable instead of the database itself still allows third-parties to construct consistent estimators of a wide class of population statistics. Second, the algorithm can satisfy the definition of differential privac…
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 requirement that the reproducing kernel is characteristic, i.e., the embedding is injective. In this paper, we generalize this embedding to finite signed Borel measures, wherein any finite signed Borel measure is represented as a mean element in an RKHS. We show that the proposed e…
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 on learning a parametrization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel em…
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 previously been shown that when the kernel used in defining the joint embedding is characteristic (that is, the embedding of the joint distribution to the feature space is injective), then the HSIC-based test is consistent. In particular, it is sufficient for the product of kerne…
This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates constraints on bound selection for the embedded domain, thus improving the robustness, as illustrated with a test case with 25 variables and intrinsic dimension 6.
Mahidol University（玛希隆大学）；Idiap Research Institute（Idiap研究所）；University of Bern（伯尔尼大学）；Centre of Excellence in Mathematics（数学卓越中心）
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 optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels ty…
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 $d_k$ over $M$. We study three questions. (I) For a given kernel, what sets $M$ can be embedded? (II) When is the embedding injective over $M$ (in which case $d_k$ is a metric)? (III) How does the $d_k$-induced topology compare to other topologies on $M$? The existing machine learning literature has addressed these questions in c…
We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function $f$, consistent estimators of the mean embedding of a random variable $X$ lead to consistent estimators of the mean embedding of $f(X)$. For Matérn kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are i…
