# Intrinsic minimum average variance estimation for sufficient dimension reduction with symmetric positive definite matrices and beyond

## TL;DR

This paper introduces intrinsic methods for sufficient dimension reduction with symmetric positive definite matrices, leveraging Riemannian geometry to improve estimation accuracy and extend applicability.

## Contribution

It develops the intrinsic minimum average variance estimation and outer product gradient methods using Riemannian metrics, extending to general manifolds and providing rigorous theoretical guarantees.

## Key findings

- Methods outperform existing techniques in simulations.
- Algorithms effectively estimate structural dimension with theoretical support.
- Application to taxi network data demonstrates practical utility.

## Abstract

In this paper, we target the problem of sufficient dimension reduction with symmetric positive definite matrices valued responses. We propose the intrinsic minimum average variance estimation method and the intrinsic outer product gradient method which fully exploit the geometric structure of the Riemannian manifold where responses lie. We present the algorithms for our newly developed methods under the log-Euclidean metric and the log-Cholesky metric. Each of the two metrics is linked to an abelian Lie group structure that transforms our model defined on a manifold into a Euclidean one. The proposed methods are then further extended to general Riemannian manifolds. We establish rigourous asymptotic results for the proposed estimators, including the rate of convergence and the asymptotic normality. We also develop a cross validation algorithm for the estimation of the structural dimension with theoretical guarantee Comprehensive simulation studies and an application to the New York taxi network data are performed to show the superiority of the proposed methods.