Constrained Matrix and Tensor Factorizations and Their Application in Data Fusion

• Applicant: Professor Dr. Max Pfeffer
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 448293816

The proposed research project aims to study constrained matrix and tensor factorizations in data fusion. The main application is the classification of cancer samples. In data fusion, several complementary measurements are taken for each sample, whereupon a joint matrix (or tensor) factorization is performed in order to reduce the dimensionality, to find relevant component (or feature) vectors, and to cluster the data points. A number of different constraints can be imposed on the factor matrices, thus improving the clustering and extracting meaningful feature vectors that can subsequently be interpreted.We aim to apply the techniques of data fusion to less well understood cancers, thus building on a previous and preparatory work. We will validate the resulting clusterings with the help of collaborations in biology and we will investigate the impact of the joint and the constrained aspect of the factorizations on the clustering. In this context, constrained tensor factorizations will be applied and studied as well.Furthermore, we will introduce nonstandard constraints into data fusion, for example sparse or even boolean or subtropical factorizations. This will help to include a wider range of data types and to improve the clustering and subsequent feature selection process. Methods will be derived that allow to select the relevant features for the clustering, especially in relation to the choice of constraints.Finally, we will derive and implement the algorithms that are necessary to perform the above computations. For this, we will combine the strength of Riemannian optimization, which is a standard tool in the computation of low rank factorizations, with that of proximal algorithms, which allow to solve nonsmooth (but convex) problems. This also has many applications outside of data fusion.

• DFG Programme: Research Grants