2014/07/07 by Kim Batselier, Batselier, Kim, Jianchun Li +3
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1407.1593
openalex publication_date 2014/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, named TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via the singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime with properties such as uniqueness for a fixed order of indices, orthogonal rank-1 outer product terms, and easy truncation error quantification. Using an outer product column table it also allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor. Incidentally, this leads to a strikingly simple constructive proof showing that the maximum rank of a real 2 × 2 × 2 tensor over the real field is 3. We also derive a conversion of the TTr1 decomposition into a Tucker decomposition with a sparse core tensor. Numerical examples illustrate each of the favorable properties of the TTr1 decomposition.