Tracking Online Low-Rank Approximations of Higher-Order Incomplete Streaming Tensors

In this paper, we propose two new provable algorithms for tracking online low-rank approximations of higher-order streaming tensors in the presence of missing data. The first algorithm, dubbed adaptive CP decomposition (ACP), minimizes an exponentially weighted recursive least-squares cost function to obtain the tensor factors in an efficient way, thanks to the alternative minimization framework and the randomized sketching technique. Under the Tucker model, the second algorithm called adaptive Tucker decomposition (ATD), which is more flexible than the first one, first tracks the underlying low-dimensional subspaces covering the tensor factors, and then estimates the core tensor using a stochastic approximation. Both algorithms are fast, and require a low computational complexity and memory storage. A unified convergence analysis is presented for ACP and ATD to justify their performance. Experiments indicate that the two proposed algorithms are capable of the adaptive tensor decomposition problem with competitive performance on both synthetic and real data.