Less is More: Compact Matrix Decomposition for Large Sparse Graphs

Abstract

Given a large sparse graph, how can we find patterns and anomalies? Several important applications can be modeled as large sparse graphs, e.g., network traffic monitoring, research citation network analysis, social network analysis, and regulatory networks in genes. Low rank decompositions, such as SVD and CUR, are powerful techniques for revealing latent/hidden variables and associated patterns from high dimensional data. However, those methods often ignore the sparsity property of the graph, and hence usually incur too high memory and computational cost to be practical.We propose a novel method, the Compact Matrix Decomposition (CMD), to compute sparse low rank approximations. CMD dramatically reduces both the computation cost and the space requirements over existing decomposition methods (SVD, CUR). Using CMD as the key building block, we further propose procedures to efficiently construct and analyze dynamic graphs from real-time application data. We provide theoretical guarantee for our methods, and present results on two real, large datasets, one on network flow data (100GB trace of 22K hosts over one month) and one on DBLP (200MB over 25 years).We show that CMD is often an order of magnitude more efficient than the state of the art (SVD and CUR): it is over 10X faster, but requires less than 1/10 of the space, for the same reconstruction accuracy. Finally, we demonstrate how CMD is used for detecting anomalies and monitoring timeevolving graphs, in which it successfully detects worm-like hierarchical scanning patterns in real network data.

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

“…We compare LS-DCUR against CUR-L2 (Euclidean-norm based selection), CUR-SL (statistical leverage based selection), and Greedy (a recent deterministic selection method that was shown to exceed various other methods [7]). To provide a fair comparison, we incorporate several extensions into the importance sampling based methods: both CUR-L2 and CUR-SL use the extensions proposed for CMD [26] and, in both cases, we sample exactly the same number of unique rows and columns as in the case of LS-DCUR and Greedy (double selections do not count as a selected row or column). For methods requiring computation of the top-k singular vectors (CUR-SL, Greedy), we specify a reasonable k. As setting it to the actual number of sampled rows and columns is not advisable, we follow the suggestion of [22] and over-sampled k; various experimental runs show that setting k to ≈ 4 5 of the number of row and column samples provides a convenient tradeoff between run-time performance and approximation accuracy; note that LS-DCUR does not require any additional parameters apart from the number of desired rows and columns.…”

“…Various extensions to this method have been proposed. For example, the approach in [26] further reduces computation times by avoiding to repeatedly sample the same row (or column).…”

“…The implicit selection criteria (large norm, large distance, orthogonality) have two interesting implications: first, the risk of multiple selections of the same sample is negligible. Thereby, similar to CMD [26], we reduce the size of the matrices C, R and decrease computation times for U (note that most available implementations of CUR already incorporate this step as did we in our experiments). Second, selection methods that apply importance sampling based on Euclidean are not applicable to matrices that consist of unit-norm vectors such as, for example, normalized Histograms.…”

“…Corresponding approaches yield interpretable results because they embed the data in lower dimensional spaces whose basis vectors correspond to actual data points. They are guaranteed to preserve properties such as sparseness or nonnegativity and enjoy increasing popularity in the data mining community [3,11,12,17,21,22,26,28] where they have been applied to fraud detection, fMRI segmentation, collaborative filtering, and co-clustering.…”

“…Approaches that rely on probability distributions which depend on the statistical leverage, i.e. the top-k singular subspace of a matrix [22], or the Euclidean norm of rows and columns [9,26] were introduced under the name CUR decomposition. As the CUR decomposition relies on informed row-and column-sampling strategies, it can scale to large data but only if statistical leverage is consider for k being rather small since the CUR algorithm involves computing the SVD of a matrix.…”