Abstract: Eigenvector Localization--the phenomenon where Graph Laplacian Eigenfunctions attain "small" values except for a neighborhood of the underlying domain--has gathered recent interest due to its applications in graph signal processing. In this talk we will discuss a slight variant of this phenomenon on Combinatorial Laplacians called Pure Localization, where an eigenfunction vanishes on a vertex and its neighborhood. We will review key definitions from spectral graph theory, and then go over some important examples of graphs which admit pure localization. We will also prove a gluing formula for purely localizing graphs. The last part of the talk will involve setting up the Neumann and Dirichlet theory of the Graph Laplacian, with the goal of proving that pure localization is equivalent to the existence of joint Dirichlet and Neumann eigenvectors. Using this equivalence, we will derive some simple necessary conditions for purely localizing graphs.
Abstract: Eigenvector Localization--the phenomenon where Graph Laplacian Eigenfunctions attain "small" values except for a neighborhood of the underlying domain--has gathered recent interest due to its applications in graph signal processing. In this talk we will discuss a slight variant of this phenomenon on Combinatorial Laplacians called Pure Localization, where an eigenfunction vanishes on a vertex and its neighborhood. We will review key definitions from spectral graph theory, and then go over some important examples of graphs which admit pure localization. We will also prove a gluing formula for purely localizing graphs. The last part of the talk will involve setting up the Neumann and Dirichlet theory of the Graph Laplacian, with the goal of proving that pure localization is equivalent to the existence of joint Dirichlet and Neumann eigenvectors. Using this equivalence, we will derive some simple necessary conditions for purely localizing graphs.
Abstract: The goal of the talk is to describe a bi-Lipshitz embedding of the quotient space of matrices modulo row permutation and of quotient space of symmetric square matrices modulo row and column permutation. In the talk we will see an example of a bi-Lipshitz embedding b_A created by sorting after multiplying a key matrix A. For a fixed matrix A, the set of matrices X, such that b_A(X)= b_A(Y) iff Y is a row permutation of Y are called the separation set of A. We will talk about properties of separated sets, and then explain under what conditions the map b_A is an Bi-Lipschitz injection of this quotient space.
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