In machine learning, high dimensional data generally should have a high degree of freedom. However, recent experiments in machine learning show that real world data in high dimensions is usually governed by a surprisingly low dimensions. We believe that in high dimensions, geometry information, for example, the “shape” of data distribution, can help learning algorithms to perform better. A geom...

A flip or edge-replacement can be considered as a transformation by which one edge e of a geometric figure is removed and an edge f (f 6= e) is inserted such that the resulting figure belongs to the same class as the original figure. This thesis is concerned with transformation of two kinds of geometric objects, namely, planar trees and planar paths, through the application of flips. A techniqu...

We introduce multiscale covariance fields associated with probability measures on Euclidean space and use them to define local scales at a point and to construct shape transforms. Local scales at x may be interpreted as scales at which key geometric features of data organization around x are revealed. Shape transforms are used to identify points that are most salient in terms of the local-globa...

Abstract The Distributed Video Coding (DVC) paradigm is based on two well-known information theory results: the SlepianWolf and the Wyner-Ziv theorems. In a DVC codec, the correlation between video signals is exploited at the decoder, providing a flexible distribution of the computational complexity between the encoder and the decoder and error robustness to the channel errors. To exploit the c...