Multilinear singular value decomposition of a tensor with fibers observed along one mode*

Abstract

We introduce an algorithm that uses only standard linear algebra operations for computing the multilinear singular value decomposition of an incomplete tensor with fibers observed along a single mode. This setting is very relevant for applications. For example, in an application where the tensor has a "time" mode, obtaining a fiber along this mode may be considerably easier than doing so along other modes. In the noise-free case, the algorithm is guaranteed to retrieve the exact solution, if the observed fibers satisfy certain deterministic conditions. As such, the approach reveals an interesting feature of the tensor setting that is not present at the matrix level. In the presence of noise, a solution obtained with this algorithm serves as a good initial point for further optimization. We illustrate, both on synthetic and real-life data, that this initialization strategy is fast and significantly reduces the number of iterations needed by an optimization algorithm. One possible use of the approach is as a linear algebra-based orthogonal compression of an incomplete tensor, after which the low multilinear rank approximation can be used as a "complete" proxy of the data for further analysis.