Exact Tensor Completion Powered by Slim Transforms
Li Ge, Lin Chen, Yudong Chen, Xue Jiang
TL;DR
This work advances exact tensor completion by enabling recovery with arbitrary linear transforms through a transform-domain tensor framework. By operating directly in the transform domain, it defines a transform-domain tensor algebra (including ⊛, t-SVD, and tensor norms) and proves a high-probability exact recovery guarantee under incoherence, even for non-orthogonal transforms. A key insight is that slim, non-invertible transforms can outperform square transforms, a result supported by an energy-based analysis and ensemble transform design. The authors validate the theory with extensive experiments on random tensors and visual data (MSI and video), showing superior reconstruction and inpainting performance, thereby broadening the practical applicability of tensor completion. The approach also accommodates data-driven and framelet-like transforms, expanding the toolkit for tensor recovery in multiway data applications.
Abstract
In this work, a tensor completion problem is studied, which aims to perfectly recover the tensor from partial observations. The existing theoretical guarantee requires the involved transform to be orthogonal, which hinders its applications. In this paper, jumping out of the constraints of isotropy and self-adjointness, the theoretical guarantee of exact tensor completion with arbitrary linear transforms is established by directly operating the tensors in the transform domain. With the enriched choices of transforms, a new analysis obtained by the proof discloses why slim transforms outperform their square counterparts from a theoretical level. Our model and proof greatly enhance the flexibility of tensor completion and extensive experiments validate the superiority of the proposed method.