We present new adaptive sampling rules for the sketch-and-project method for solving linear systems. To deduce our new sampling rules, we first show how the progress of one step of the sketch-and-project method depends directly on a sketched residual. Based on this insight, we derive a 1) max-distance sampling rule, by sampling the sketch with the largest sketched residual 2) a proportional sampling rule, by sampling proportional to the sketched residual, and finally 3) a capped sampling rule. The capped sampling rule is a generalization of the recently introduced adaptive sampling rules for the Kaczmarz method [3]. We provide a global linear convergence theorem for each sampling rule and show that the max-distance rule enjoys the fastest convergence. This finding is also verified in extensive numerical experiments that lead us to conclude that the max-distance sampling rule is superior both experimentally and theoretically to the capped sampling rule. We also provide numerical insights into implementing the adaptive strategies so that the per iteration cost is of the same order as using a fixed sampling strategy when the number of sketches times the sketch size is not significantly larger than the number of columns.
An active area of research in computational science is the design of algorithms for solving the subgraph matching problem to find copies of a given template graph in a larger world graph. Prior works have largely addressed single-channel networks using a variety of approaches. We present a suite of filtering methods for subgraph isomorphisms for multiplex networks (with different types of edges between nodes and more than one edge within each channel type). We aim to understand the entire solution space rather than focusing on finding one isomorphism. Results are shown on several classes of datasets: (a) Sudoku puzzles mapped to the subgraph isomorphism problem, (b) Erd ős-R ényi multigraphs, (c) real-world datasets from Twitter and transportation networks, (d) synthetic data created for the DARPA MAA program.
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