Incremental tensor analysis: Theory and applications

Abstract

How do we find patterns in author-keyword associations, evolving over time? Or in data cubes (tensors), with product-branchcustomer sales information? And more generally, how to summarize high-order data cubes (tensors)? How to incrementally update these patterns over time? Matrix decompositions, like principal component analysis (PCA) and variants, are invaluable tools for mining, dimensionality reduction, feature selection, rule identification in numerous settings like streaming data, text, graphs, social networks, and many more settings. However, they have only two orders (i.e., matrices, like author and keyword in the previous example). We propose to envision such higher-order data as tensors, and tap the vast literature on the topic. However, these methods do not necessarily scale up, let alone operate on semi-infinite streams. Thus, we introduce a general framework, incremental tensor analysis (ITA), which efficiently computes a compact summary for high-order and high-dimensional data, and also reveals the hidden correlations. Three variants of ITA are presented: (1) dynamic tensor analysis (DTA); (2) streaming tensor analysis (STA); and (3) window-based tensor analysis (WTA). In paricular, we explore several fundamental design trade-offs such as space efficiency, computational cost, approximation accuracy, time dependency, and model complexity. We implement all our methods and apply them in several real settings, such as network anomaly detection, multiway latent semantic indexing on citation networks, and correlation study on sensor measurements. Our empirical studies show that the proposed methods are fast and accurate and that they find interesting patterns and outliers on the real datasets.