Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.Comment: To appear in FOCS 201

S2: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S2 for this task. At each step, S2 selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S2 queries for the label of the vertex that bisects the *shortest shortest* path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S2 needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S2 on both real and synthetic data. While other graph-based active learning algorithms have shown promise in practice, our algorithm is the first with both good performance and theoretical guarantees. Finally, we demonstrate the implications of the S2 algorithm to the theory of nonparametric active learning. In particular, we show that S2 achieves near minimax optimal excess risk for an important class of nonparametric classification problems.Comment: A version of this paper appears in the Conference on Learning Theory (COLT) 201

Minimizing Hitting Time between Disparate Groups with Shortcut Edges

Structural bias or segregation of networks refers to situations where two or more disparate groups are present in the network, so that the groups are highly connected internally, but loosely connected to each other. In many cases it is of interest to increase the connectivity of disparate groups so as to, e.g., minimize social friction, or expose individuals to diverse viewpoints. A commonly-used mechanism for increasing the network connectivity is to add edge shortcuts between pairs of nodes. In many applications of interest, edge shortcuts typically translate to recommendations, e.g., what video to watch, or what news article to read next. The problem of reducing structural bias or segregation via edge shortcuts has recently been studied in the literature, and random walks have been an essential tool for modeling navigation and connectivity in the underlying networks. Existing methods, however, either do not offer approximation guarantees, or engineer the objective so that it satisfies certain desirable properties that simplify the optimization~task. In this paper we address the problem of adding a given number of shortcut edges in the network so as to directly minimize the average hitting time and the maximum hitting time between two disparate groups. Our algorithm for minimizing average hitting time is a greedy bicriteria that relies on supermodularity. In contrast, maximum hitting time is not supermodular. Despite, we develop an approximation algorithm for that objective as well, by leveraging connections with average hitting time and the asymmetric k-center problem.Comment: To appear in KDD 202

Análise de desempenho de métricas de grafos para reconhecimento de tarefas de imaginação motora das mãos a partir de dados de eletroencefalografia

Orientadores: Gabriela Castellano, Romis Ribeiro de Faissol AttuxDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb WataghinResumo: Interfaces cérebro-computador (BCIs, brain-computer interfaces) são sistemas cuja finalidade é fornecer um canal de comunicação direto entre o cérebro e um dispositivo externo, como um computador, uma prótese ou uma cadeira de rodas. Por não utilizarem as vias fisiológicas convencionais, BCIs podem constituir importantes tecnologias assistivas para pessoas que sofreram algum tipo de lesão e, por isso, tiveram sua interação com o ambiente externo comprometida. Os sinais cerebrais a serem extraídos para utilização nestes sistemas devem ser gerados mediante estratégias específicas. Nesta dissertação, trabalhamos com a estratégia de imaginação motora (MI, motor imagery), e extraímos a resposta cerebral correspondente a partir de dados de eletroencefalografia (EEG). Os objetivos do trabalho foram caracterizar as redes cerebrais funcionais oriundas das tarefas de MI das mãos e explorar a viabilidade de utilizar métricas da teoria de grafos para a classificação dos padrões mentais, gerados por esta estratégia, de usuários de um sistema BCI. Para isto, fez-se a hipótese de que as alterações no espectro de frequências dos sinais de eletroencefalografia devidas à MI das mãos deveria, de alguma forma, se refletir nos grafos construídos para representar as interações cerebrais corticais durante estas tarefas. Em termos de classificação, diferentes conjuntos de pares de eletrodos foram testados, assim como diferentes classificadores (análise de discriminantes lineares ¿ LDA, máquina de vetores de suporte ¿ SVM ¿ linear e polinomial). Os três classificadores testados tiveram desempenho similar na maioria dos casos. A taxa média de classificação para todos os voluntários considerando a melhor combinação de eletrodos e classificador foi de 78%, sendo que alguns voluntários tiveram taxas de acerto individuais de até 92%. Ainda assim, a metodologia empregada até o momento possui várias limitações, sendo a principal como encontrar os pares ótimos de eletrodos, que variam entre voluntários e aquisições; além do problema da realização online da análiseAbstract: Brain-computer interfaces (BCIs) are systems that aim to provide a direct communication channel between the brain and an external device, such as a computer, a prosthesis or a wheelchair. Since BCIs do not use the conventional physiological pathways, they can constitute important assistive technologies for people with lesions that compromised their interaction with the external environment. Brain signals to be extracted for these systems must be generated according to specific strategies. In this dissertation, we worked with the motor imagery (MI) strategy, and we extracted the corresponding cerebral response from electroencephalography (EEG) data. Our goals were to characterize the functional brain networks originating from hands¿ MI and investigate the feasibility of using metrics from graph theory for the classification of mental patterns, generated by this strategy, of BCI users. We hypothesized that frequency alterations in the EEG spectra due to MI should reflect themselves, in some manner, in the graphs representing cortical interactions during these tasks. For data classification, different sets of electrode pairs were tested, as well as different classifiers (linear discriminant analysis ¿ LDA, and both linear and polynomial support vector machines ¿ SVMs). All three classifiers tested performed similarly in most cases. The mean classification rate over subjects, considering the best electrode set and classifier, was 78%, while some subjects achieved individual hit rates of up to 92%. Still, the employed methodology has yet some limitations, being the main one how to find the optimum electrode pairs¿ sets, which vary among subjects and among acquisitions; in addition to the problem of performing an online analysisMestradoFísicaMestre em Física165742/2014-31423625/2014CNPQCAPE

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails