Persistent Homology in Sparse Regression and its Application to Brain Morphometry
This work addresses a fundamental bottleneck in sparse regression for researchers in machine learning and neuroscience, though it is incremental as it applies existing topological methods to a known problem.
The paper tackles the problem of selecting the tuning parameter in sparse regression by introducing persistent homology over the parameter space, resulting in a soft-thresholding technique that speeds up computation and reveals that stress-exposed children exhibit more diffuse white matter organization in brain morphometry.
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.