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圖的拉普拉斯矩陣學習-Laplacian Matrices of Graphs

We all learn one way of solving linear equations when we first encounter linear algebra: Gaussian Elimination. In this survey, I will tell the story of some remarkable connections between algorithms, spectral graph theory, functional analysis and numerical linear algebra that arise in the search for asymptotically faster algorithms. I will only consider the problem of solving systems of linear equations in the Laplacian matrices of graphs. This is a very special case, but it is also a very interesting case. I begin by introducing the main characters in the story. 1. Laplacian Matrices and Graphs. We will consider weighted, undirected, simple graphs G given by a triple (V,E,w), where V is a set of vertices, E is a set of edges, and w is a weight function that assigns a positive weight to every edge. The Laplacian matrix L of a graph is most naturally defined by the quadratic form it induces. For a vector x ∈ IRV , the Laplacian quadratic form of G is: Thus, L provides a measure of the smoothness of x over the edges in G. The more x jumps over an edge, the larger the quadratic form becomes. The Laplacian L also has a simple description as a matrix. Define the weighted degree of a vertex u by: Define D to be the diagonal matrix whose diagonal contains d, and define the weighted adjacency matrix of G by:
We have L = D − A. It is often convenient to consider the normalized Laplacian of a graph instead of the Laplacian. It is given by D−1/2LD−1/2, and is more closely related to the behavior of random walks. Regression on Graphs. Imagine that you have been told the value of a function f on a subset W of the vertices of G, and wish to estimate the values of f at the remaining vertices. Of course, this is not possible unless f respects the graph structure in some way. One reasonable assumption is that the quadratic form in the Laplacian is small, in which case one may estimate f by solving for the function f : V → IR minimizing f TLf subject to f taking the given values on W (see [ZGL03]). Alternatively, one could assume that the value of f at every vertex v is the weighted average of f at the neighbors of v, with the weights being proportional to the edge weights. In this case, one should minimize: |D-1
Lf| subject to f taking the given values on W. These problems inspire many uses of graph Laplacians in Machine Learning.