Low-Rank Matrix and Tensor Completion via Adaptive Sampling
This addresses the problem of efficient data reconstruction from incomplete observations for researchers in machine learning and signal processing, offering incremental improvements over prior methods.
The paper tackles low-rank matrix and tensor completion by proposing adaptive sampling algorithms that achieve exact recovery from fewer entries, specifically Ω(n r^{3/2} log(r)) for matrices and Ω(n r^{T-1/2} T^2 log(r)) for tensors, even under high coherence conditions.
We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results hold even when the row space is highly coherent, in contrast with previous analyses. In the absence of noise, we show that one can exactly recover a $n \times n$ matrix of rank $r$ from merely $Ω(n r^{3/2}\log(r))$ matrix entries. We also show that one can recover an order $T$ tensor using $Ω(n r^{T-1/2}T^2 \log(r))$ entries. For noisy recovery, our algorithm consistently estimates a low rank matrix corrupted with noise using $Ω(n r^{3/2} \textrm{polylog}(n))$ entries. We complement our study with simulations that verify our theory and demonstrate the scalability of our algorithms.