|Abstract: ||High-dimensional high-order arrays, or tensors, arise in many modern scientific applications including genomics, brain imaging, and social science. In this talk, we consider two specific problems in tensor data analysis: low-rank tensor completion and tensor PCA.
We first propose a framework for low-rank tensor completion via a novel tensor measurement scheme we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that the tensors of Tucker low-rank can be recovered from a limited number of noiseless measurements, which matches the sample complexity lower-bound. In the case of noisy measurements, we also develop a theoretical upper bound and the matching minimax lower bound for recovery error over certain classes of low-rank tensors for the proposed procedure. Finally, the procedure is illustrated through a real dataset in neuroimaging.
Then we propose a general framework of tensor principal component analysis. The problem exhibits different phases based on signal-noise-ratio (SNR): with strong SNR, we propose an efficient power method that achieves the minimax optimal rate in estimation error in sine Theta norms; with weak SNR, we provide information theoretical lower bound to show this problem is statistically impossible; with moderate SNR, there is a trade-off between statistical accuracy and computational complexity.|