2015/02/12 by Dehua Cheng, Yu Cheng, Cheng, Dehua +8 · 2 citations
Computer Science · Mathematics · #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Matrix Theory and Algorithms #Random Matrices and Applications #Social and Information Networks (cs.SI) #Topological and Geometric Data Analysis #cs.DM #cs.DS #cs.LG #cs.SI #stat.ML
paper · pdf · doi:10.48550/arxiv.1502.03496
arxiv created 2015/02/12 · openalex publication_date 2015/02/12 · arxiv updated 2015/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial Lα(G)=D-∑r=1dαrD(D-1A)r where A is the adjacency matrix of a weighted, undirected graph, D is the diagonal matrix of weighted degrees, and α=(α1...αd) are nonnegative coefficients with ∑r=1dαr=1. Recall that D-1A is the transition matrix of random walks on the graph. The sparsification of Lα(G) appears to be algorithmically challenging as the matrix power (D-1A)r is defined by all paths of length r, whose precise calculation would be prohibitively expensive. In this paper, we develop the first nearly linear time algorithm for this sparsification problem: For any G with n vertices and m edges, d coefficients α, and ε> 0, our algorithm runs in time O(d2mlog2n/ε2) to construct a Laplacian matrix L=D-A with O(nlog n/ε2) non-zeros such that L≈εLα(G). Matrix polynomials arise in mathematical analysis of matrix functions as well as numerical solutions of matrix equations. Our work is particularly motivated by the algorithmic problems for speeding up the classic Newton's method in applications such as computing the inverse square-root of the precision matrix of a Gaussian random field, as well as computing the qth-root transition (for q≥1) in a time-reversible Markov model. The key algorithmic step for both applications is the construction of a spectral sparsifier of a constant degree random-walk matrix-polynomials introduced by Newton's method. Our algorithm can also be used to build efficient data structures for effective resistances for multi-step time-reversible Markov models, and we anticipate that it could be useful for other tasks in network analysis.