Graph inference and graph matching problems : tehory and algorithms

Tribunal: Alex Bronstein (Tel Aviv University), Marcelo Lanzilotta (Universidad de la República), Gonzalo Mateos (University of Rochester), Gadiel Seroussi (Universidad de la República)Almost every field has some problems related with graphs or networks. From natural examples in physics and mathematics, to applications in medicine and signal processing, graphs are either a very powerful tool, or a very rich object of interest. In this thesis we address two classes of graph-related problems. First, we focus on graph-inference problems, consisting in the estimation of a graph or network from a dataset. In this part of the manuscript, we modify the existing formulations of the inference problem to incorporate prior topological information of the graph, and to jointly infer several graphs in a collaborative way. We apply these techniques to infer genetic regulation networks, brain connectivity patterns, and economyrelated networks. We also present a new problem, which consists of the estimation of mobility patterns from highly asynchronous and incomplete data. We give a first formulation of the problem with its corresponding optimization, and present results for airplane routes and New York taxis mobility patterns. The second class consists of the so-called graph matching problems. In this type of problems two graphs are given, and the objective is to find the best alignment between them. This problem is of great interest both from an algorithmic and theoretical point of view, besides the very important applications. Its interest and difficulty lie in the combinatorial nature of the problem: the cost of seeking among all the possible permutations grows exponentially with the number of nodes, and hence becomes intractable even for small graphs. First, we focus on the algorithmic aspect of the graph matching problem. We present two methods based on relaxations of the discrete optimization problem. The first one is inspired in ideas from the sparse modeling community, and the second one is based on a theorem presented in this manuscript. The importance of these methods is illustrated with several applications. Finally, we address some theoretical aspects about graph matching and other related problems. The main question tackled in the last chapter is the following: when do the graph matching problem and its convex relaxation have the same solution? A probabilistic approach is first given, showing that, asymptotically, the most common convex relaxation fails, while a non-convex relaxation succeeds with probability one if the graphs to be matched are correlated enough, showing a phase-transition type of behavior. On the other hand, a deterministic approach is presented, stating conditions on the eigenvectors and eigenvalues of the adjacency matrix for guaranteeing the correctness of the convex relaxation solution. Other results and conjectures relating the spectrum and symmetry of a graph are presented as well.En prácticamente todos los campos hay problemas relacionados con grafos o redes. Desde los ejemplos más naturales en física y matem ática, hasta aplicaciones en medicina y procesamiento de señales, los grafos son una herramienta muy poderosa, o un objeto de estudio muy rico e interesante. En esta tesis atacamos dos clases de problemas relacionados con grafos. Primero, nos enfocamos en problemas de inferencia de grafos, que consisten en estimar un grafo o red a partir de cierto conjunto de datos. En esta parte del manuscrito, modificamos las formulaciones existentes de inferencia de grafos para incorporar información topológica previamente conocida sobre el grafo, y para inferir de manera conjunta varios grafos, en un modo colaborativo. Aplicamos estas técnicas para inferir redes de regulaci ón genética, patrones de conectividad cerebral, y redes relacionadas con el mercado accionario. También presentamos un nuevo problema, que consiste en la estimación de patrones de movimiento a partir de un conjunto de datos incompleto, y altamente asíncrono. Mostramos primero una formulación del problema con su correspondiente optimización, y presentamos resultados para rutas de aviones en Estados Unidos, y patrones de movilidad de taxis en New York. La segunda clase consiste en los llamados graph matching problems (problemas de apareamiento de grafos). En este tipo de problemas, dos grafos son dados, y el objetivo es encontrar el mejor alineamiento entre ellos. Este problema es de gran interés tanto desde un punto de vista algorítmico como teórico, además de las importantes aplicaciones que tiene. El interés y la dificultad de este problema tienen raíz en la naturaleza combinatoriadel mismo: el costo de buscar entre todas las permutaciones posibles crece exponencialmente con el número de nodos, y por lo tanto se vuelve rápidamente intratable, incluso para grafos chicos. Primero, nos enfocamos en el aspecto algorítmico del problema de graph match- ing. Presentamos dos métodos basados en relajaciones del problema de optimización discreta. El primero de ellos está inspirado en ideas de la comunidad de sparse modeling, y el segundo est a basado en un teorema presentado en este manuscritp. La importancia de estos m etodos es ilustrada con varias aplicaciones a lo largo del capítulo. Finalmente, atacamos algunos aspectos teóricos sobre graph matching y otros problemas relacionados. La pregunta principal que se encara en el último capítulo es la siguiente: >cuáando el problema de graph matching y su relajación convexa tienen la misma solucióon? Primero damos un enfoque probabilístico mostrando que, asintoticamente, la relajación convexa más común falla, mientras que una relajación no convexa es capaz de resolver el problema con probabilidad uno, siempre y cuando los grafos originales estén lo sufi cientemente correlacionados, mostrando un comportamiento del estilo de transicióon de fases. Por otro lado, un enfoque determinístico es también presentado, estableciendo condiciones sobre los valores y vectores propios de las matrices de adjacencia de los grafos, que garantizan que el problema de graph matching y su relajacióon convexa tienen la misma solución. Otros resultados y conjeturas relacionando el espectro y la simetría de un grafo son presentados también en este capítulo

Segmentation and polyp detection in virtual colonoscopy : a complete system for computer aided diagnosis

El cancer colorectal es una de las mayores causas de muerte por cancer en el mundo. La deteccion temprana de polipos es fundamental para su tratamiento, permitiendo alcanzar tasas del 90% de curabilidad. La tecnica habitual para la deteccion de polipos, debido a su elevada performance, es la colonoscopia optica (tecnica invasiva y extremadamente cara). A mediados de los '90 surge la tecnica denominada colonoscopia virtual. Esta tecnica consiste en la reconstruccion 3D del colon a partir de cortes de tomografia computada. Es por ende una tecnica no invasiva, y relativamente barata, pero la cantidad de falsos positivos y falsos negativos producida por estos metodos esta muy por encima de los maximos aceptados en la practica medica. Los avances recientes en las tecnicas de imagenologia parecerian hacer posible la reduccion de estas tasas. Como consecuencia de esto, estamos asistiendo a un nuevo interes por la colonoscopia virtual. En este trabajo se presenta un sistema completo de diagnostico asistido por computadora. La primera etapa del sistema es la segmentacion, que consiste en la reconstruccion 3D de la superficie del colon a partir del volumen tomografico. El aporte principal en este paso es el suavizado de la imagen. A partir de la superficie, se detectan aquellas zonas candidatas de ser polipos mediante una estrategia multi-escala que permite delinear con precision la zona. Luego para cada candidato se extraen caracteristicas geometricas y de textura, que son calculadas tambien en el tejido que rodea la zona a efectos de compararlas. Finalmente las zonas candidatas se clasifican utilizando SVM. Los resultados obtenidos son prometedores, permitiendo detectar un 100% de los polipos mayoresColorectal cancer is the second leading cause of cancer-related death in the United States, and the third cause worldwide. The early detection of polyps is fundamental, allowing to reduce mortality rates up to 90%. Nowadays, optical colonoscopy is the most used detection method due in part to its relative high performance. Virtual Colonoscopy is a promising alternative technique that emerged in the 90's. It uses volumetric Computed Tomographic data of the cleansed and air-distended colon, and the examination is made by a specialist from the images in a computer. Therefore, this technique is less invasive and less expensive than optical colonoscopy, but up to now the false positive and false negative rates are above the accepted medical limits. Recent advances in imaging techniques have the potential to reduce these rates; consequently, we are currently re-experiencing an increasing interest in Virtual Colonoscopy. In this work we propose a complete pipeline for a Computer-Aided Detection algorithm. The system starts with a novel and simple segmentation step. We then introduce geometrical and textural features that take into account not only the candidate polyp region, but the surrounding area at multiple scales as well. This way, our proposed CAD algorithm is able to accurately detect candidate polyps by measuring local variations of these features. Candidate patches are then classi ed using SVM. The whole algorithm is completely automatic and produces state-of-the-art results, achieving 100% sensitivity for polyps greater than 6mm in size with less than one false positive per case, and 100% sensitivity for polyps greater than 3mm in size with 2:2 false positives per case

Online change point detection for weighted and directed random dot product graphs

Given a sequence of random (directed and weighted) graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. Our idea is to endow sequential change-point detection (CPD) techniques with a graph representation learning substrate based on the versatile Random Dot Product Graph (RDPG) model. We consider efficient, online updates of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal RDPG. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence. We characterize the distribution of this running statistic to select thresholds that guarantee error-rate control, and under simplifying approximations we offer insights on the algorithm’s detection resolution and delay. The end result is a lightweight online CPD algorithm, that is also explainable by virtue of the well-appreciated interpretability of RDPG embeddings. This is in stark contrast with most existing graph CPD approaches, which either rely on extensive computation, or they store and process the entire observed time series. An apparent limitation of the RDPG model is its suitability for undirected and unweighted graphs only, a gap we aim to close here to broaden the scope of the CPD framework. Unlike previous proposals, our non-parametric RDPG model for weighted graphs does not require a priori specification of the weights’ distribution to perform inference and estimation. This network modeling contribution is of independent interest beyond CPD. We offer an open-source implementation of the novel online CPD algorithm for weighted and direct graphs, whose effectiveness and efficiency are demonstrated via (reproducible) synthetic and real network data experimentsWork in this paper is supported in part by ANII (grant FMV 3 2018 1 148149) and the NSF (awards CCF-1750428, CCF-1934962 and ECCS-1809356). Part of the results in this paper were submitted to the 2021 EUSIPCO and Asilomar Conference

Decoupling between SARS-CoV-2 transmissibility and population mobility associated with increasing immunity from vaccination and infection in South America

All South American countries from the Southern cone (Argentina, Brazil, Chile, Paraguay and Uruguay) experienced severe COVID-19 epidemic waves during early 2021 driven by the expansion of variants Gamma and Lambda, however, there was an improvement in different epidemic indicators since June 2021. To investigate the impact of national vaccination programs and natural infection on viral transmission in those South American countries, we analyzed the coupling between population mobility and the viral effective reproduction number Rt. Our analyses reveal that population mobility was highly correlated with viral Rt from January to May 2021 in all countries analyzed; but a clear decoupling occurred since May–June 2021, when the rate of viral spread started to be lower than expected from the levels of social interactions. These findings support that populations from the South American Southern cone probably achieved the conditional herd immunity threshold to contain the spread of regional SARS-CoV-2 variants circulating at that time