Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

Cluster and Feature Modeling from Combinatorial Stochastic Processes

One of the focal points of the modern literature on Bayesian nonparametrics has been the problem of clustering, or partitioning, where each data point is modeled as being associated with one and only one of some collection of groups called clusters or partition blocks. Underlying these Bayesian nonparametric models are a set of interrelated stochastic processes, most notably the Dirichlet process and the Chinese restaurant process. In this paper we provide a formal development of an analogous problem, called feature modeling, for associating data points with arbitrary nonnegative integer numbers of groups, now called features or topics. We review the existing combinatorial stochastic process representations for the clustering problem and develop analogous representations for the feature modeling problem. These representations include the beta process and the Indian buffet process as well as new representations that provide insight into the connections between these processes. We thereby bring the same level of completeness to the treatment of Bayesian nonparametric feature modeling that has previously been achieved for Bayesian nonparametric clustering.Comment: Published in at http://dx.doi.org/10.1214/13-STS434 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org