Faculty of Engineering and Natural Sciences
Spherical-Homoscedastic Distributions and the Design of Bayes Optimal Classifiers
Dept. of Electrical and Computer Engineering,
The Ohio State University, Columbus, OH, USA
Many feature representations, as in genomics, describe directional data where all feature vectors share a common norm. In other cases, as in computer vision, a norm or variance normalization step, where all feature vectors are normalized to a common length, is generally used. These representations and pre-processing step map the original data from a Euclidean space to the surface of a hypersphere. Such representations should then be modeled using spherical distributions. However, the difficulty associated with such spherical representations has impaired their use in practice. Instead, researchers commonly model their spherical data using Gaussian distributions -- as if the data were represented in Rp rather than Sp-1. This opens the question to whether the classification results calculated with the Gaussian approximation are the same as those obtained when using the original spherical distributions. In this talk, I will show that in some particular cases (which we have named spherical-homoscedastic) the answer to this question is positive. In the more general case, however, the answer is negative. I will then show that the more the data deviates from spherical-homoscedastic, the less advisable it is to employ the Gaussian approximation. This then lends to the derivation of a set of Bayes optimal classifiers for spherical-homoscedastic and –heteroscedastic distributions -- the latter being derived by means of an appropriate kernel function that maps the original feature space into one where the data adapts to the spherical-homoscedastic model. The resulting non-linear classifiers have applications in a large number of real problems, of which, I will show examples in classifying images of objects, gene expression sequences, and text data. Time permitting I will sketch the problem of Bayes optimality in feature extraction.
July 6, 2007, 11:00, FENS G032