Spectrally approximating large graphs with smaller graphs

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification.Comment: 22 pages, 10 figure

How to Round Subspaces: A New Spectral Clustering Algorithm

A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different basis? In this paper, we propose a new spectral clustering algorithm. It can recover a $k$-partition such that the subspace corresponding to the span of its indicator vectors is $O(\sqrt{opt})$ close to the original subspace in spectral norm with $opt$ being the minimum possible ($opt \le 1$ always). Moreover our algorithm does not impose any restriction on the cluster sizes. Previously, no algorithm was known which could find a $k$-partition closer than $o(k \cdot opt)$. We present two applications for our algorithm. First one finds a disjoint union of bounded degree expanders which approximate a given graph in spectral norm. The second one is for approximating the sparsest $k$-partition in a graph where each cluster have expansion at most $\phi_k$ provided $\phi_k \le O(\lambda_{k+1})$ where $\lambda_{k+1}$ is the $(k+1)^{st}$ eigenvalue of Laplacian matrix. This significantly improves upon the previous algorithms, which required $\phi_k \le O(\lambda_{k+1}/k)$.Comment: Appeared in SODA 201

Efficient Bounds and Estimates for Canonical Angles in Randomized Subspace Approximations

Randomized subspace approximation with "matrix sketching" is an effective approach for constructing approximate partial singular value decompositions (SVDs) of large matrices. The performance of such techniques has been extensively analyzed, and very precise estimates on the distribution of the residual errors have been derived. However, our understanding of the accuracy of the computed singular vectors (measured in terms of the canonical angles between the spaces spanned by the exact and the computed singular vectors, respectively) remains relatively limited. In this work, we present bounds and estimates for canonical angles of randomized subspace approximation that can be computed efficiently either a priori or a posterior. Under moderate oversampling in the randomized SVD, our prior probabilistic bounds are asymptotically tight and can be computed efficiently, while bringing a clear insight into the balance between oversampling and power iterations given a fixed budget on the number of matrix-vector multiplications. The numerical experiments demonstrate the empirical effectiveness of these canonical angle bounds and estimates on different matrices under various algorithmic choices for the randomized SVD

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE