Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups

09/18/2020
by   Zhenhua Lin, et al.
0

In this paper an additive regression model for a symmetric positive-definite matrix valued response and multiple scalar predictors is proposed. The model exploits the abelian group structure inherited from either the Log-Cholesky metric or the Log-Euclidean framework that turns the space of symmetric positive-definite matrices into a Riemannian manifold and further a bi-invariant Lie group. The additive model for responses in the space of symmetric positive-definite matrices with either of these metrics is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions but also allows to generalize the proposed additive model to general Riemannian manifolds that might not have a Lie group structure. Optimal asymptotic convergence rates and normality of the estimated component functions are also established. Numerical studies show that the proposed model enjoys superior numerical performance, especially when there are multiple predictors. The practical merits of the proposed model are demonstrated by analyzing diffusion tensor brain imaging data.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 14

08/25/2019

Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition

We present a new Riemannian metric, termed Log-Cholesky metric, on the m...
02/01/2021

Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices

In this paper, we develop a new classification method for manifold-value...
06/24/2020

Modelling the Statistics of Cyclic Activities by Trajectory Analysis on the Manifold of Positive-Semi-Definite Matrices

In this paper, a model is presented to extract statistical summaries to ...
05/14/2018

H-CNNs: Convolutional Neural Networks for Riemannian Homogeneous Spaces

Convolutional neural networks are ubiquitous in Machine Learning applica...
07/15/2020

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Rie-ma...
08/27/2018

Intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

This paper develops intrinsic wavelet denoising methods for surfaces of ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.