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Topology-based kernels with application to inference problems in Alzheimer's disease.

Alzheimer’s Disease Neuroimaging Initiative and Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI 53706, USA.
IEEE transactions on medical imaging 04/2011; 30(10):1760-70. DOI: 10.1109/TMI.2011.2147327
Source: PubMed

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 K between input examples (i.e., images). The inner product between images I(i) and I(j) in a feature space can generally be written in closed form and so it is convenient to treat K 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 Alzheimer's Disease Neuroimaging Initiative study, we report good performance on several statistical inference tasks without any feature selection, dimensionality reduction, or parameter tuning.

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    • "With the same advances in modern computing technology that allow for the storage of large datasets, persistent homology and its variants can be implemented. Features derived from persistent homology have recently been found useful for classification of hepatic lesions (Adcock et al., 2014) and persistent homology has been applied for the analysis of structural brain images (Chung et al., 2009; Pachauri et al., 2011). Outside the arena of medical applications, Sethares and Budney (2013) use persistent homology to study topological structures in musical data. "
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    • "Yet, the step of training the classifier with topological information is typically done in a rather adhoc manner. In [23] for instance, the persistence diagram is first rasterized on a regular grid, then a kernel-density estimate is computed, and eventually the vectorized discrete probability density function is used as a feature vector to train a SVM using standard kernels for R n . It is however unclear how the resulting kernel-induced distance behaves with respect to existing metrics (e.g., bottleneck or Wasserstein distance) and how properties such as stability are affected. "
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