Many classification algorithms produce a classifier that is a weighted average of kernel evaluations. When working with a high or infinite dimensional kernel, it is imperative for speed of evaluation and storage issues that as few training samples as possible are used in the kernel expansion. Popular existing approaches focus on altering standard learning algorithms, such as the Support Vector Machine, to induce sparsity, as well as post-hoc procedures for sparse approximations. Here we adopt the latter approach. We begin with a very simple classifier, given by the kernel mean $$ f(x) = \frac{1}{n} \sum\limits_{i=i}^{n} y_i K(x_i,x) $$ We then find a sparse approximation to this kernel mean via herding. The result is an accurate, easily parallelized algorithm for learning classifiers.

Feature Learning aims to extract relevant information contained in data sets in an automated fashion. It is driving force behind the current deep learning trend, a set of methods that have had widespread empirical success. What is lacking is a theoretical understanding of different feature learning schemes. This work provides a theoretical framework for feature learning and then characterizes when features can be learnt in an unsupervised fashion. We also provide means to judge the quality of features via rate-distortion theory and its generalizations.

"Deep Learning" methods attempt to learn generic features in an unsupervised fashion from a large unlabelled data set. These generic features should perform as well as the best hand crafted features for any learning problem that makes use of this data. We provide a definition of generic features, characterize when it is possible to learn them and provide methods closely related to the autoencoder and deep belief network of deep learning. In order to do so we use the notion of deficiency and illustrate its value in studying certain general learning problems.