A Statistical Recurrent Model On The Manifold Of Symmetric Positive Definite Matrices

Authors:
Rudrasis Chakraborty University of Florida
Chun-Hao Yang University of Florida
Xingjian Zhen UW-Madison
Monami Banerjee University of Florida
Derek Archer University of Florida
David Vaillancourt University of Florida
Vikas Singh UW-Madison
Baba Vemuri University of Florida, USA

Introduction:

In a number of disciplines, the data (e.g., graphs, manifolds) to beanalyzed are non-Euclidean in nature.In this work, the authors study the setting where the data(or measurements) are ordered, longitudinal or temporal in nature andlive on a Riemannian manifold -- this setting is common in a varietyof problems in statistical machine learning, vision and medicalimaging.

Abstract:

In a number of disciplines, the data (e.g., graphs, manifolds) to beanalyzed are non-Euclidean in nature. Geometric deep learningcorresponds to techniques that generalize deep neural network modelsto such non-Euclidean spaces. Several recent papers have shown howconvolutional neural networks (CNNs) can be extended to learn withgraph-based data. In this work, we study the setting where the data(or measurements) are ordered, longitudinal or temporal in nature andlive on a Riemannian manifold -- this setting is common in a varietyof problems in statistical machine learning, vision and medicalimaging. We show how recurrent statistical recurrent network modelscan be defined in such spaces. We give an efficient algorithm andconduct a rigorous analysis of its statistical properties. We performextensive numerical experiments demonstrating competitive performancewith state of the art methods but with significantly less number ofparameters. We also show applications to a statistical analysis taskin brain imaging, a regime where deep neural network models have onlybeen utilized in limited ways.

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