Mahalanobis Distance on Extended Grassmann Manifolds for Variational Pattern Analysis

In pattern classification problems, pattern variations are often modeled as a linear manifold or a low-dimensional subspace. Conventional methods use such models and define a measure of similarity or dissimilarity. However, these similarity measures are deterministic and do not take into account the distribution of linear manifolds or low-dimensional subspaces. Therefore, if the distribution is not isotopic, the distance measurements are not reliable, as well as vector-based distance measurement in the Euclidean space. We previously systematized the representations of variational patterns using the Grassmann manifold and introduce the Mahalanobis distance to the Grassmann manifold as a natural extension of Euclidean case. In this paper, we present two methods that flexibly extend the Mahalanobis distance on the extended Grassmann manifolds. These methods can be used to measure pattern (dis)similarity on the basis of the pattern structure. Experimental evaluation of the performance of the proposed methods demonstrated that they exhibit a lower error classification rate.