Exact heat kernel on a hypersphere and its applications in kernel SVM

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed, demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This paper presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis

Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysis

Motor imagery (MI) classification is one of the most widely-concern research topics in Electroencephalography (EEG)-based brain-computer interfaces (BCIs) with extensive industry value. The MI-EEG classifiers' tendency has changed fundamentally over the past twenty years, while classifiers' performance is gradually increasing. In particular, owing to the need for characterizing signals' non-Euclidean inherence, the first geometric deep learning (GDL) framework, Tensor-CSPNet, has recently emerged in the BCI study. In essence, Tensor-CSPNet is a deep learning-based classifier on the second-order statistics of EEGs. In contrast to the first-order statistics, using these second-order statistics is the classical treatment of EEG signals, and the discriminative information contained in these second-order statistics is adequate for MI-EEG classification. In this study, we present another GDL classifier for MI-EEG classification called Graph-CSPNet, using graph-based techniques to simultaneously characterize the EEG signals in both the time and frequency domains. It is realized from the perspective of the time-frequency analysis that profoundly influences signal processing and BCI studies. Contrary to Tensor-CSPNet, the architecture of Graph-CSPNet is further simplified with more flexibility to cope with variable time-frequency resolution for signal segmentation to capture the localized fluctuations. In the experiments, Graph-CSPNet is evaluated on subject-specific scenarios from two well-used MI-EEG datasets and produces near-optimal classification accuracies.Comment: 16 pages, 5 figures, 9 Tables; This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Forman-Ricci flow for change detection in large dynamic data sets

We present a viable solution to the challenging question of change detection in complex networks inferred from large dynamic data sets. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci flow that already proved to be a powerful tool in image processing and graphics, to the case of undirected and weighted networks. The application of the proposed method on peer-to-peer networks yields insights into topological properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version

Investigating Brain Functional Networks in a Riemannian Framework

The brain is a complex system of several interconnected components which can be categorized at different Spatio-temporal levels, evaluate the physical connections and the corresponding functionalities. To study brain connectivity at the macroscale, Magnetic Resonance Imaging (MRI) technique in all the different modalities has been exemplified to be an important tool. In particular, functional MRI (fMRI) enables to record the brain activity either at rest or in different conditions of cognitive task and assist in mapping the functional connectivity of the brain. The information of brain functional connectivity extracted from fMRI images can be defined using a graph representation, i.e. a mathematical object consisting of nodes, the brain regions, and edges, the link between regions. With this representation, novel insights have emerged about understanding brain connectivity and providing evidence that the brain networks are not randomly linked. Indeed, the brain network represents a small-world structure, with several different properties of segregation and integration that are accountable for specific functions and mental conditions. Moreover, network analysis enables to recognize and analyze patterns of brain functional connectivity characterizing a group of subjects. In recent decades, many developments have been made to understand the functioning of the human brain and many issues, related to the biological and the methodological perspective, are still need to be addressed. For example, sub-modular brain organization is still under debate, since it is necessary to understand how the brain is functionally organized. At the same time a comprehensive organization of functional connectivity is mostly unknown and also the dynamical reorganization of functional connectivity is appearing as a new frontier for analyzing brain dynamics. Moreover, the recognition of functional connectivity patterns in patients affected by mental disorders is still a challenging task, making plausible the development of new tools to solve them. Indeed, in this dissertation, we proposed novel methodological approaches to answer some of these biological and neuroscientific questions. We have investigated methods for analyzing and detecting heritability in twin's task-induced functional connectivity profiles. in this approach we are proposing a geodesic metric-based method for the estimation of similarity between functional connectivity, taking into account the manifold related properties of symmetric and positive definite matrices. Moreover, we also proposed a computational framework for classification and discrimination of brain connectivity graphs between healthy and pathological subjects affected by mental disorder, using geodesic metric-based clustering of brain graphs on manifold space. Within the same framework, we also propose an approach based on the dictionary learning method to encode the high dimensional connectivity data into a vectorial representation which is useful for classification and determining regions of brain graphs responsible for this segregation. We also propose an effective way to analyze the dynamical functional connectivity, building a similarity representation of fMRI dynamic functional connectivity states, exploiting modular properties of graph laplacians, geodesic clustering, and manifold learning

Interdisciplinary and physics challenges of Network Theory

Network theory has unveiled the underlying structure of complex systems such as the Internet or the biological networks in the cell. It has identified universal properties of complex networks, and the interplay between their structure and dynamics. After almost twenty years of the field, new challenges lie ahead. These challenges concern the multilayer structure of most of the networks, the formulation of a network geometry and topology, and the development of a quantum theory of networks. Making progress on these aspects of network theory can open new venues to address interdisciplinary and physics challenges including progress on brain dynamics, new insights into quantum technologies, and quantum gravity.Comment: (7 pages, 4 figures