A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CN...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-10
Main Authors: Chakraborty, Rudrasis, Chun-Hao, Yang, Xingjian Zhen, Banerjee, Monami, Archer, Derek, Vaillancourt, David, Singh, Vikas, Vemuri, Baba C
Format: Article
Language:English
Subjects:
Algorithms
Artificial neural networks
Brain
Euclidean geometry
Machine learning
Manifolds (mathematics)
Mathematical models
Matrix methods
Medical imaging
Neural networks
Riemann manifold
Statistical analysis
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended to learn with graph-based data. In this work, we study the setting where the data (or measurements) are ordered, longitudinal or temporal in nature and live on a Riemannian manifold -- this setting is common in a variety of problems in statistical machine learning, vision and medical imaging. We show how recurrent statistical recurrent network models can be defined in such spaces. We give an efficient algorithm and conduct a rigorous analysis of its statistical properties. We perform extensive numerical experiments demonstrating competitive performance with state of the art methods but with significantly less number of parameters. We also show applications to a statistical analysis task in brain imaging, a regime where deep neural network models have only been utilized in limited ways.
ISSN:2331-8422