Joint Tensor Factorization and Outlying Slab Suppression with Applications

We consider factoring low-rank tensors in the presence of outlying slabs. This problem is important in practice, because data collected in many real-world applications, such as speech, fluorescence, and some social network data, fit this paradigm. Prior work tackles this problem by iteratively selecting a fixed number of slabs and fitting, a procedure which may not converge. We formulate this problem from a group-sparsity promoting point of view, and propose an alternating optimization framework to handle the corresponding $\ell_p$ ($0<p\leq 1$) minimization-based low-rank tensor factorization problem. The proposed algorithm features a similar per-iteration complexity as the plain trilinear alternating least squares (TALS) algorithm. Convergence of the proposed algorithm is also easy to analyze under the framework of alternating optimization and its variants. In addition, regularization and constraints can be easily incorporated to make use of \emph{a priori} information on the latent loading factors. Simulations and real data experiments on blind speech separation, fluorescence data analysis, and social network mining are used to showcase the effectiveness of the proposed algorithm

Nonlinear System Identification via Tensor Completion

Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general nonlinear system, as they have proven to be very effective in approximating complex highly nonlinear functions. In this work, we show that identifying a general nonlinear function $y = f(x_1,\ldots,x_N)$ from input-output examples can be formulated as a tensor completion problem and under certain conditions provably correct nonlinear system identification is possible. Specifically, we model the interactions between the $N$ input variables and the scalar output of a system by a single $N$-way tensor, and setup a weighted low-rank tensor completion problem with smoothness regularization which we tackle using a block coordinate descent algorithm. We extend our method to the multi-output setting and the case of partially observed data, which cannot be readily handled by neural networks. Finally, we demonstrate the effectiveness of the approach using several regression tasks including some standard benchmarks and a challenging student grade prediction task.Comment: AAAI 202