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.

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.

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.