[Paper Review] Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups

Zhenhua Lin, Hans‐Georg Müller+1 more|arXiv (Cornell University)|Sep 18, 2020

Advanced Neuroimaging Techniques and Applications29 references4 citations

TL;DR

This paper proposes a novel additive regression model for symmetric positive-definite (SPD) matrix-valued responses using Riemannian manifold structures, specifically leveraging the Log-Cholesky and Log-Euclidean metrics to transform the nonlinear SPD space into a linear tangent space. The method enables efficient estimation via smooth backfitting, achieves optimal convergence rates of $O_P(n^{-4/5})$ for component function estimation, and establishes asymptotic normality, significantly improving performance over full nonparametric methods in high-dimensional predictor settings.

ABSTRACT

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.

Motivation & Objective

• To address the lack of additive regression models for symmetric positive-definite (SPD) matrix-valued responses in high-dimensional predictor settings.
• To overcome the curse of dimensionality inherent in full nonparametric regression methods like local polynomial regression for SPD data.
• To establish a theoretically grounded framework that connects SPD-valued additive models to additive models on tangent spaces via Riemannian geometry.
• To generalize the proposed model to broader Riemannian manifolds beyond Lie groups by exploiting metric and group structure.
• To demonstrate practical utility through application to diffusion tensor imaging (DTI) data, particularly in neuroscience.

Proposed method

• The model uses the Log-Cholesky or Log-Euclidean metric to map the SPD manifold into a Riemannian manifold with abelian group and bi-invariant Lie group structure.
• It transforms the SPD-valued response into the tangent space at a mean matrix via the logarithmic map, enabling additive modeling in a linear space.
• The additive component functions are estimated using smooth backfitting on the transformed tangent space responses, with constraints ensuring identifiability.
• Parallel transport is used to map estimated functions back to the original manifold, preserving geometric consistency.
• The method leverages the framework of Jeon and Park (2020) for Hilbert-space-valued additive models, adapted to tangent spaces of Riemannian manifolds.
• Asymptotic theory is developed using functional central limit theorems, with convergence rates derived under regularity conditions on the design and error structure.

Experimental results

Research questions

• RQ1 Can additive regression be effectively extended to symmetric positive-definite matrix-valued responses while avoiding the curse of dimensionality?
• RQ2 How can the geometric structure of the SPD manifold—specifically its Riemannian and Lie group properties—be exploited to enable efficient, nonparametric estimation?
• RQ3 What are the optimal convergence rates and asymptotic distributional properties of the estimated component functions in this setting?
• RQ4 Can the proposed method be generalized beyond SPD matrices to arbitrary Riemannian manifolds lacking a Lie group structure?
• RQ5 How does the model perform in practice compared to existing nonparametric regression methods on real-world neuroimaging data?

Key findings

• The proposed additive model achieves optimal convergence rates of $O_P(n^{-4/5})$ for estimating component functions under the Log-Cholesky or Log-Euclidean metric.
• The model ensures asymptotic normality of the estimated component functions, with joint convergence to a normal distribution under appropriate bandwidth scaling.
• The method outperforms full nonparametric regression in high-dimensional predictor settings, particularly when multiple predictors are present.
• The transformation to the tangent space via the logarithmic map enables efficient smooth backfitting, reducing computational complexity and improving numerical stability.
• Theoretical guarantees are extended to general Riemannian manifolds by exploiting the tangent space connection, even without a Lie group structure.
• Numerical studies and application to Alzheimer’s Disease Neuroimaging Initiative (ADNI) DTI data confirm superior performance and practical relevance in neuroscience applications.