Optimal low rank tensor recovery
This provides a computationally efficient method for tensor completion, which is incremental but has practical applications in areas like hyperspectral image processing.
The paper tackles the problem of recovering a low-rank tensor from a small subset of its entries, showing that a Riemannian optimization algorithm with spectral initialization can achieve exact recovery with high probability using as few as O((r^d + dnr) log(d)) entries for a tensor of size n^d and rank r, with O(nr) entries for order-3 tensors.
We investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. In the Tucker decomposition framework, we show that the Riemannian optimization algorithm with initial value obtained from a spectral method can reconstruct a tensor of size $n\times n \times\cdots \times n$ tensor of ranks $(r,\cdots,r)$ with high probability from as few as $O((r^d+dnr)\log(d))$ entries. In the case of order 3 tensor, the entries can be asymptotically as few as $O(nr)$ for a low rank large tensor. We show the theoretical guarantee condition for the recovery. The analysis relies on the tensor restricted isometry property (tensor RIP) and the curvature of the low rank tensor manifold. Our algorithm is computationally efficient and easy to implement. Numerical results verify that the algorithms are able to recover a low rank tensor from minimum number of measurements. The experiments on hyperspectral images recovery also show that our algorithm is capable of real world signal processing problems.