Topology-Based Kernels With Application to Inference Problems in Alzheimer's Disease

Abstract

Alzheimer’s disease (AD) research has recently witnessed a great deal of activity focused on developing new statistical learning tools for automated inference using imaging data. The workhorse for many of these techniques is the Support Vector Machine (SVM) framework (or more generally kernel based methods). Most of these require, as a first step, specification of a kernel matrix between input examples (i.e., images). The inner product between images Ii and Ij in a feature space can generally be written in closed form, and so it is convenient to treat as “given”. However, in certain neuroimaging applications such an assumption becomes problematic. As an example, it is rather challenging to provide a scalar measure of similarity between two instances of highly attributed data such as cortical thickness measures on cortical surfaces. Note that cortical thickness is known to be discriminative for neurological disorders, so leveraging such information in an inference framework, especially within a multi-modal method, is potentially advantageous. But despite being clinically meaningful, relatively few works have successfully exploited this measure for classification or regression. Motivated by these applications, our paper presents novel techniques to compute similarity matrices for such topologically-based attributed data. Our ideas leverage recent developments to characterize signals (e.g., cortical thickness) motivated by the persistence of their topological features, leading to a scheme for simple constructions of kernel matrices. As a proof of principle, on a dataset of 356 subjects from the ADNI study, we report good performance on several statistical inference tasks without any feature selection, dimensionality reduction, or parameter tuning.

Dissimilarity data in statistical model building and machine learning

“…[18]. Nonnegative definite similarity matrices have in recent years been frequently used in statistical model building for images; for a recent example see [31]. A recent discussion of the use of noisy (indefinite) similarity matrices and methods for transforming them to kernel matrices can be found in [5].…”

Persistent homology for the quantitative prediction of fullerene stability

“…Numerous software packages, such as Perseus, Dionysus, and Javaplex, 47 based on various algorithms have been developed and made available in the public domain. As an efficient tool to unveil topological invariants, persistent homology has been applied to various fields, such as image analysis, 5, 40, 45 chaotic dynamics verification, 30, 36 sensor network, 44 complex network, 29, 34 data analysis, 4 geometric processing, 16 and computational biology. 10, 22, 31, 55 Based on persistent homology analysis, we have proposed molecular topological fingerprints and utilized them to reveal the topology-function relationship of biomolecules.…”

Multiresolution persistent homology for excessively large biomolecular datasets

“…Many elegant computational algorithms have been proposed for persistent homology analysis in the literature. 6,7,10,21 There is a long list of successful applications of persistent homology in a variety of fields, including data analysis, 4,19,22,26,31 image analysis, 3,5,12,25,29 shape recognition, 8 chaotic dynamics verification, 16,20 network structure, 15,18,28 computer vision, 29 and computational biology. 13,17,39 Topological characterization identification and analysis (CIA) are some of the most successful applications of persistent homology.…”