Geometric structure of sum-of-rank-1 decompositions for n-dimensional order-p symmetric tensors

The canonical sum-of-rank-one decomposition of tensors is a fundamental linear algebraic problem encountered in signal processing, machine learning, and other scientific fields. Current algorithms that emerge from CANDECOMP or PARAFAC formalisms rely on the basic definition of tensor decomposition that describes rank as the minimum number of vectors that are needed to reconstruct the tensor using outer product linear combinations, which is an extension of the same property of matrix rank. In this paper, we reinterpret the orthogonality condition of symmetric matrix eigenvectors as a geometric constraint on the coordinate frame formed by the eigenvectors and relaxing the orthogonality, we develop a set of structured-bases that can be utilized to decompose any symmetric tensor into its sum-of-rank-one (canonical) decomposition. The eigenvectors of order-p tensors are observed to form a frame where the angle between various pairs of eigenvectors are integer multiples of pi/p. Validation of the proposed geometric structure and demonstration of decomposition accuracies obtained using these frames (at the level of a computer´s numerical-epsiv) are provided.