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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Available from: Deepti Pachauri, Jul 31, 2014
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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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    ABSTRACT: Brain decoding involves the determination of a subject's cognitive state or an associated stimulus from functional neuroimaging data measuring brain activity. In this setting the cognitive state is typically characterized by an element of a finite set, and the neuroimaging data comprise voluminous amounts of spatiotemporal data measuring some aspect of the neural signal. The associated statistical problem is one of classification from high-dimensional data. We explore the use of functional principal component analysis, mutual information networks, and persistent homology for examining the data through exploratory analysis and for constructing features characterizing the neural signal for brain decoding. We review each approach from this perspective, and we incorporate the features into a classifier based on symmetric multinomial logistic regression with elastic net regularization. The approaches are illustrated in an application where the task is to infer, from brain activity measured with magnetoencephalography (MEG), the type of video stimulus shown to a subject.
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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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    ABSTRACT: Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.
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    • "Since the AD-specific brain changes begin years before the patient becomes symptomatic, early clinical diagnosis becomes a challenging task. Accordingly, there have been a lot of studies focusing on possible identification of such changes at the early stage, i.e., mild cognitive impairment (MCI), by leveraging neuroimaging data [2], [3]. Recently, machine learning and pattern classification approaches have been widely used to identify AD and MCI at an individual level [4]–[8], rather than at a group level, i.e., only the comparison between different clinical groups. "
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    ABSTRACT: Rapid advances in neuroimaging techniques have provided an efficient and non-invasive way for exploring the structural and functional connectivity of the human brain. Quantitative measurement of abnormality of brain connectivity in patients with neurodegenerative diseases, such as mild cognitive impairment (MCI) and Alzheimer's disease (AD), have also been widely reported, especially at a group level. Recently, machine learning techniques have been applied to the study of AD and MCI, i.e., to identify the individuals with AD/MCI from the healthy controls (HC). However, most existing methods focus on using only a single property of a connectivity network, although multiple network properties, such as local connectivity and global topological properties, can potentially be used. In this paper, by employing multi-kernel based approach, we propose a novel connectivity based framework to integrate multiple properties of connectivity network for improving the classification performance. Specifically, two different types of kernels (i.e., vector-based kernel and graph kernel) are used to quantify two different yet complementary properties of the network, i.e., local connectivity and global topological properties. Then, multi-kernel learning (MKL) technique is adopted to fuse these heterogeneous kernels for neuroimaging classification. We test the performance of our proposed method on two different data sets. First, we test it on the functional connectivity networks of 12 MCI and 25 HC subjects. The results show that our method achieves significant performance improvement over those using only one type of network property. Specifically, our method achieves a classification accuracy of 91.9%, which is 10.8% better than those by single network-property based methods. Then, we test our method for gender classification on a large set of functional connectivity networks with 133 infants scanned at birth, 1-year-old, and 2-years-old, also demonstrating very promising results.
    IEEE transactions on bio-medical engineering 10/2013; 61(2). DOI:10.1109/TBME.2013.2284195 · 2.23 Impact Factor
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