Simplicial variances, potentials and Mahalanobis distances

Learning Local Invariant Mahalanobis Distances

For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, and tolerance for image transformations was primarily achieved by using robust...

Envelope plots of ordered Mahalanobis distances: uses and efficient generation

Systematic departures from a reference sampling scheme, such as multivariate normality, can affect all the observed ordered Mahalanobis distances (OMDs) – not just a few extreme values – causing distinctive changes to their overall pattern. Envelope OMD plots provide a useful framework against which to ‘read’ such changes, often suggestive of further, confirmatory analyses. Illustrative example...

Lipschitz Continuity of Mahalanobis Distances and Bilinear Forms

Many theoretical results in the machine learning domain stand only for functions that are Lipschitz continuous. Lipschitz continuity is a strong form of continuity that linearly bounds the variations of a function. In this paper, we derive tight Lipschitz constants for two families of metrics: Mahalanobis distances and bounded-space bilinear forms. To our knowledge , this is the first time the ...

Moments, Factor Scores and Limiting Distributions of Individual Mahalanobis Distances

Fuzzy C-Means Algorithm Based on Standard Mahalanobis Distances

Some of the well-known fuzzy clustering algorithms are based on Euclidean distance function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, the former needs added constraint of fuzzy covariance matrix, the later can only be used for the d...