Ryan O'Dowd

H. N. Mhaskar, Ryan O'Dowd

A fundamental problem in manifold learning is to approximate a functional relationship in a data chosen randomly from a probability distribution supported on a low dimensional sub-manifold of a high dimensional ambient Euclidean space. The manifold is essentially defined by the data set itself and, typically, designed so that the data is dense on the manifold in some sense. The notion of a data space is an abstraction of a manifold encapsulating the essential properties that allow for function approximation. The problem of transfer learning (meta-learning) is to use the learning of a function on one data set to learn a similar function on a new data set. In terms of function approximation, this means lifting a function on one data space (the base data space) to another (the target data space). This viewpoint enables us to connect some inverse problems in applied mathematics (such as inverse Radon transform) with transfer learning. In this paper we examine the question of such lifting when the data is assumed to be known only on a part of the base data space. We are interested in determining subsets of the target data space on which the lifting can be defined, and how the local smoothness of the function and its lifting are related.

H. N. Mhaskar, Ryan O'Dowd

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively ``rough'' functions.

Hrushikesh Mhaskar, Ryan O'Dowd, Efstratios Tsoukanis

In machine learning, classification is usually seen as a function approximation problem, where the goal is to learn a function that maps input features to class labels. In this paper, we propose a novel clustering and classification framework inspired by the principles of signal separation. This approach enables efficient identification of class supports, even in the presence of overlapping distributions. We validate our method on real-world hyperspectral datasets Salinas and Indian Pines. The experimental results demonstrate that our method is competitive with the state of the art active learning algorithms by using a very small subset of data set as training points.

Ryan O'Dowd

Machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before. This dissertation focuses on three different projects rooted in mathematical theory for machine learning applications. The first project deals with supervised learning and manifold learning. In theory, one of the main problems in supervised learning is that of function approximation: that is, given some data set $\mathcal{D}=\{(x_j,f(x_j))\}_{j=1}^M$, can one build a model $F\approx f$? We introduce a method which aims to remedy several of the theoretical shortcomings of the current paradigm for supervised learning. The second project deals with transfer learning, which is the study of how an approximation process or model learned on one domain can be leveraged to improve the approximation on another domain. We study such liftings of functions when the data is assumed to be known only on a part of the whole domain. We are interested in determining subsets of the target data space on which the lifting can be defined, and how the local smoothness of the function and its lifting are related. The third project is concerned with the classification task in machine learning, particularly in the active learning paradigm. Classification has often been treated as an approximation problem as well, but we propose an alternative approach leveraging techniques originally introduced for signal separation problems. We introduce theory to unify signal separation with classification and a new algorithm which yields competitive accuracy to other recent active learning algorithms while providing results much faster.

H. N. Mhaskar, Ryan O'Dowd

The problem of classification in machine learning has often been approached in terms of function approximation. In this paper, we propose an alternative approach for classification in arbitrary compact metric spaces which, in theory, yields both the number of classes, and a perfect classification using a minimal number of queried labels. Our approach uses localized trigonometric polynomial kernels initially developed for the point source signal separation problem in signal processing. Rather than point sources, we argue that the various classes come from different probability distributions. The localized kernel technique developed for separating point sources is then shown to separate the supports of these distributions. This is done in a hierarchical manner in our MASC algorithm to accommodate touching/overlapping class boundaries. We illustrate our theory on several simulated and real life datasets, including the Salinas and Indian Pines hyperspectral datasets and a document dataset.