Optimization of Block-based Tensor Decompositions through Sub-Tensor Impact Graphs and Applications to Dynamicity in Data and User Focus

Tensors are commonly used for representing multi-dimensional data, such as Web graphs, sensor streams, and social networks. As a consequence of the increase in the use of tensors, tensor decomposition operations began to form the basis for many data analysis and knowledge discovery tasks, from clustering, trend detection, anomaly detection to correlationanalysis [31, 38]. It is well known that Singular Value matrix Decomposition (SVD) [9] is used to extract latent semantics for matrix data. When apply SVD to tensors, which have more than two modes, it is tensor decomposition. The two most popular tensor decomposition algorithms are the Tucker [54] and the CP [19] decompositions. Intuitively, they both generalize SVD to tensors. However, one key problem with tensor decomposition is its computational complexity which may cause system bottleneck. Therefore, two phase block-centric CP tensor decomposition (2PCP) was proposed to partition the tensor into small sub-tensors, execute sub-tensor decomposition in parallel and combine the factors from each sub-tensor into final decomposition factors through iterative rerefinement process. Consequently, I proposed Sub-tensor Impact Graph (SIG) to account for inaccuracy propagation among sub-tensors and measure the impact of decomposition of sub-tensors on the other's decomposition, Based on SIG, I proposed several optimization strategies to optimize 2PCP's phase-2 refinement process. Furthermore, I applied SIG and optimization strategies for data focus, data evolution, and focus shifting in tensor analysis. Personalized Tensor Decomposition (PTD) is proposed to account for the users focus given the observations that in many applications, the user may have a focus of interest i.e., part of the data for which the user needs high accuracy and beyond this area focus, accuracy may not be as critical. PTD takes as input one or more areas of focus and performs the decomposition in such a way that, when reconstructed, the accuracy of the tensor is boosted for these areas of focus. A related challenge of data evolution in tensor analytics is incremental tensor decomposition since re-computation of the whole tensor decomposition with each update will cause high computational costs and incur large memory overheads. Especially for applications where data evolves over time and the tensor-based analysis results need to be continuouslymaintained. To avoid re-decomposition, I propose a two-phase block-incremental CP-based tensor decomposition technique, BICP, that efficiently and effectively maintains tensor decomposition results in the presence of dynamically evolving tensor data. I further extend the research focus on user focus shift. User focus may change over time as data is evolving along the time. Although PTD is efficient, re-computation for each user preference update can be the bottleneck for the system. Therefore I propose dynamic evolving user focus tensor decomposition which can smartly reuse the existing decomposition result to improve the efficiency of evolving user focus block decomposition.