Optimal Recovery of Tensor Slices

We consider the problem of large scale matrix recovery given side information in the form of additional matrices of conforming dimension. This is a parsimonious model that captures a number of interesting problems including context and location aware recommendations, personalized ‘tag’ learning, demand learning with side information, etc. Viewing the matrix we seek to recover and the side information we have as slices of a tensor, we consider the problem of Slice Recovery, which is to recover specific slices of a tensor from noisy observations of the tensor. We provide an efficient algorithm to recover slices of structurally ’simple’ tensors given noisy observations of the tensor’s entries; our definition of simplicity subsumes low-rank tensors for a variety of definitions of tensor rank. Our algorithm is practical for large datasets and provides a significant performance improvement over state of the art incumbent approaches to tensor recovery. We establish theoretical recovery guarantees that under reasonable assumptions are minimax optimal for slice recovery. These guarantees also imply the first minimax optimal guarantees for recovering tensors of low Tucker rank and general noise. Experiments on data from a music streaming service demonstrate the performance and scalability of our algorithm.