Spherical-Homoscedastic Distributions: The Equivalency of Spherical and Normal Distributions in Classification
Onur C. Hamsici, Aleix M. Martinez.
Year: 2007, Volume: 8, Issue: 56, Pages: 1583−1623
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 ℜp to the surface of a hypersphere Sp-1. Such representations should then be modeled using spherical distributions. However, the difficulty associated with such spherical representations has prompted researchers to model their spherical data using Gaussian distributions instead---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 paper, we show that in some particular cases (which we named spherical-homoscedastic) the answer to this question is positive. In the more general case however, the answer is negative. For this reason, we further investigate the additional error added by the Gaussian modeling. We conclude that the more the data deviates from spherical-homoscedastic, the less advisable it is to employ the Gaussian approximation. We then show how our derivations can be used to define optimal classifiers for spherical-homoscedastic distributions. By using a kernel which maps the original space into one where the data adapts to the spherical-homoscedastic model, we can derive non-linear classifiers with potential applications in a large number of problems. We conclude this paper by demonstrating the uses of spherical-homoscedasticity in the classification of images of objects, gene expression sequences, and text data.