Exact Tensor Completion Beyond Isotropy and Invertibility

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 invertibility for the first time, 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, we theoretically disclose why slim transforms outperform their square counterparts, providing support for existing works on experimental excellence of slim transforms. Our model and analysis greatly enhance the flexibility of tensor completion, supported by extensive experimental results.