This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
This document presents an overview of optimization algorithms on Riemannian manifolds. It begins by introducing concepts such as vector transport and retraction mappings that are used to generalize algorithms from Euclidean spaces to manifolds. It then summarizes several classical optimization methods including gradient descent, conjugate gradient, and variants of quasi-Newton methods adapted to the Riemannian setting using these geometric concepts. The convergence of the Fletcher-Reeves method is analyzed under standard assumptions on the objective function. Overall, the document provides a conceptual and mathematical foundation for optimization on manifolds.
This document discusses pattern formation in crowd dynamics. It begins with an introduction to crowd dynamics and then discusses two specific patterns: lane formation and freezing-by-heating transition. Lane formation occurs when pedestrians walking in opposite directions spontaneously form lanes to allow for more efficient movement. Freezing-by-heating transition refers to the phenomenon where increasing noise or energy in a crowd leads to the formation of orderly lanes, rather than disorder. The document explores mathematical modeling of these patterns using particle simulation models.
This document summarizes a presentation on rigorously verifying the accuracy of numerical solutions to semi-linear parabolic partial differential equations using analytic semigroups. It introduces the considered problem of finding the solution to a semi-linear parabolic PDE. It then discusses using a piecewise linear finite element discretization in space and time to obtain an initial numerical solution. The goal is to rigorously enclose the true solution within a radius ρ of this numerical solution in the function space L∞(J;H10(Ω)). Key steps involve using properties of the analytic semigroup generated by the operator A and estimating discretization errors to compute the enclosure radius ρ.
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.