Persistent Homology in Sparse Regression and its Application to Brain Morphometry

TL;DR

This paper introduces a topological approach using persistent homology to improve sparse regression analysis and applies it to brain morphometry, revealing structural differences in stress-affected children.

Contribution

It presents a novel method integrating persistent homology with sparse regression and demonstrates its application in brain morphometry for stress-related white matter analysis.

Findings

Stress-exposed children show more diffuse white matter organization.

Topological features improve sparse regression efficiency.

Method enhances multivariate brain structure analysis.

Abstract

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.