Worst-Case Completion of Tensors with Approximately Few ANOVA Terms

Abstract

In this article, the problem of completing a tensor from some incomplete knowledge of its entries is treated by adopting a worst-case perspective, given the realistic assumption that the tensor's low-order ANOVA terms are dominant. We survey and leverage some recent all-purpose results from the field of Optimal Recovery to provide solutions on a theoretical level. But the accompanying constructions of optimal completion procedures, which often feature semidefinite programs, are not directly applicable in the tensor case due to the huge dimensions involved. To resolve the issue, we put forward a storage-friendly way to produce low-order ANOVA projections based on the fast Fourier transform (FFT), while exploiting the specificities of the completion problem to efficiently compute regularizers and extremal eigenvalues. Numerical experiments on synthetic tensors and real-world datasets demonstrate the accuracy and scalability of our FFT-based method.