Tensor Networks for Latent Variable Analysis: Novel Algorithms for Tensor Train Approximation.

Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model that represents data as an ordered network of subtensors of order-2 or order-3 has, so far, not been widely considered in these fields, although this so-called tensor network (TN) decomposition has been long studied in quantum physics and scientific computing. In this article, we present novel algorithms and applications of TN decompositions, with a particular focus on the tensor train (TT) decomposition and its variants. The novel algorithms developed for the TT decomposition update, in an alternating way, one or several core tensors at each iteration and exhibit enhanced mathematical tractability and scalability for large-scale data tensors. For rigor, the cases of the given ranks, given approximation error, and the given error bound are all considered. The proposed algorithms provide well-balanced TT-decompositions and are tested in the classic paradigms of blind source separation from a single mixture, denoising, and feature extraction, achieving superior performance over the widely used truncated algorithms for TT decomposition.

Data Fusion Techniques for the Integration of Multi-Domain Genomic Data from Uveal Melanoma.

Uveal melanoma (UM) is a rare cancer that is well characterized at the molecular level. Two to four classes have been identified by the analyses of gene expression (mRNA, ncRNA), DNA copy number, DNA-methylation and somatic mutations yet no factual integration of these data has been reported. We therefore applied novel algorithms for data fusion, joint Singular Value Decomposition (jSVD) and joint Constrained Matrix Factorization (jCMF), as well as similarity network fusion (SNF), for the integration of gene expression, methylation and copy number data that we applied to the Cancer Genome Atlas (TCGA) UM dataset. Variant features that most strongly impact on definition of classes were extracted for biological interpretation of the classes. Data fusion allows for the identification of the two to four classes previously described. Not all of these classes are evident at all levels indicating that integrative analyses add to genomic discrimination power. The classes are also characterized by different frequencies of somatic mutations in putative driver genes (GNAQ, GNA11, SF3B1, BAP1). Innovative data fusion techniques confirm, as expected, the existence of two main types of uveal melanoma mainly characterized by copy number alterations. Subtypes were also confirmed but are somewhat less defined. Data fusion allows for real integration of multi-domain genomic data.

On the interconnection between the higher-order singular values of real tensors.

A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted. While it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other, numerical experiments suggest that sufficiently small, but otherwise arbitrary perturbations preserve feasibility. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations, it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal, Tucker-type matricizations, but, in contrast to the matrix case, not necessary. An explicit example of this phenomenon is given.