Abstract
- In this paper, the geometry of decision border between affine sub-spaces is investigated. Affine sub-spaces are used as prototypes in machine learning approaches such as “Tangent Learning Vector Quantization" and “Tangent Distance Kernel for Support Vector Machines" for classification of data. These models assume that there are class invariant manifolds that can be locally approximated by an affine space of similar dimensions. However, in practice this assumption may not always be true, because the affine spaces compete to provide a suitable local metric that leads to proper decision boundaries for an optimal separation and classification in the feature space. Therefore, considering affine spaces together with the corresponding decision borders is necessary when drawing conclusion about the geometry of the classification problem. An understanding of the type of decision border, between two affine sub-spaces, can be used to modify related learning methods, prevent undesirable scenarios, and gain insights about the geometry of the data set. We will show that the decision borders, that are basically quadratic surfaces, can be affine spaces, Hyper-Cones, or hyperbolic paraboloids embedded in the feature space. Each type of border suggests a relative formation of data points. We will also show when a linear decision border happens.