eOTD: An Efficient Online Tucker Decomposition for Higher Order Tensors

A tensor (i.e., an N-mode array) is a natural representation for multidimensional data. Tucker Decomposition (TD) is one of the most popular methods, and a series of batch TD algorithms have been extensively studied and widely applied in signal/image processing, bioinformatics, etc. However, in many applications, the large-scale tensor is dynamically evolving at all modes, which poses significant challenges for existing approaches to track the TD for such dynamic tensors. In this paper, we propose an efficient Online Tucker Decomposition (eOTD) approach to track the TD of dynamic tensors with an arbitrary number of modes. We first propose corollaries on the multiplication of block tensor matrix. Based on this corollary, eOTD allows us 1) to update the projection matrices using those projection matrices from the previous timestamp and the auxiliary matrices from the current timestamp, and 2) to update the core tensor by a sum of tensors that are obtained by multiplying smaller tensors with matrices. The auxiliary matrices are obtained by solving a series of least square regression tasks, not by performing Singular Value Decompositions (SVD). This overcomes the bottleneck in computation and storage caused by computing SVDs on largescale data. A Modified Gram-Schmidt (MGS) process is further applied to orthonormalize the projection matrices. Theoretically, the output of the eOTD framework is guaranteed to be lowrank. We further prove that the MGS process will not increase Tucker decomposition error. Empirically, we demonstrate that the proposed eOTD achieves comparable accuracy with a significant speedup on both synthetic and real data, where the speedup can be more than 1,500 times on large-scale data.