Spectral gap-based deterministic tensor completion
This provides theoretical improvements for tensor completion problems in domains like recommender systems, though it is incremental with limitations on algorithmic efficiency.
The paper tackles tensor completion with deterministic sampling patterns by bounding generalization error for Poisson loss and atomic norm minimization methods, improving rank dependence from r^{2(t-1)(t^2-t-1)} to r^{2(t-1)(3t-5)} for order-t tensors with CP-rank r. It also reduces rank dependence for atomic tensor norm from r^{3t-3} to r^{3t-5} under random sampling, with error controlled by the spectral gap of the sampling pattern.
Tensor completion is a core machine learning algorithm used in recommender systems and other domains with missing data. While the matrix case is well-understood, theoretical results for tensor problems are limited, particularly when the sampling patterns are deterministic. Here we bound the generalization error of the solutions of two tensor completion methods, Poisson loss and atomic norm minimization, providing tighter bounds in terms of the target tensor rank. If the ground-truth tensor is order $t$ with CP-rank $r$, the dependence on $r$ is improved from $r^{2(t-1)(t^2-t-1)}$ in arXiv:1910.10692 to $r^{2(t-1)(3t-5)}$. The error in our bounds is deterministically controlled by the spectral gap of the sampling sparsity pattern. We also prove several new properties for the atomic tensor norm, reducing the rank dependence from $r^{3t-3}$ in arXiv:1711.04965 to $r^{3t-5}$ under random sampling schemes. A limitation is that atomic norm minimization, while theoretically interesting, leads to inefficient algorithms. However, numerical experiments illustrate the dependence of the reconstruction error on the spectral gap for the practical max-quasinorm, ridge penalty, and Poisson loss minimization algorithms. This view through the spectral gap is a promising window for further study of tensor algorithms.