Event Date
SPEAKER: Naoki Saito, Professor, Dept of Mathematics, UC Davis
TITLE: “The First Step Toward Building Natural Graph Wavelets”
ABSTRACT: For the development and theory of discrete wavelets on regular lattices in R^d, the Fourier series and transforms have played a significant role. Hence, when attempting to develop wavelet theory naturally tailored for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials, respectively. While tempting to do so, there are several fundamental problems in this viewpoint. One of them is the intricate relationship between the frequencies and the Laplacian eigenvalues. For undirected and unweighted paths (or cycles), the Laplacian eigenvectors are the discrete cosine (or Fourier) basis vectors and the corresponding eigenvalues are square of their frequencies. Consequently on those simple graphs, one can precisely develop the classical wavelets using the Littlewood-Paley theory. However, as soon as a graph becomes even slightly more complicated (e.g., a discretized thin rectangle in 2D), the situation completely changes: we cannot view the eigenvalues as a simple monotonic function of frequency anymore. Hence, the first step toward building natural graph wavelets is how to sort and organize Laplacian eigenfunctions without using the eigenvalues and to create a dual domain graph. In this talk, I will discuss this important problem further and explain my effort using Earth Mover's/Wasserstein Distance to measure natural distances between eigenfunctions followed by embedding the resulting distance matrix into an appropriate Euclidean domain. I will also discuss some related means to compute such distances and to organize the eigenvectors to form natural graph wavelets. The latter is joint work with Haotian Li (UCD) and Alex Cloninger (UCSD).
Speaker's web page: https://www.math.ucdavis.edu/~saito/
DATE: Thursday, March 7th, 4:10pm
LOCATION: MSB 1147, Colloquium Room
REFRESHMENTS: 3:30pm MSB 4110 (4th floor lounge)
STA 290 Seminar List: https://statistics.ucdavis.edu/seminars