The central research objective of the proposed project is the development of new, efficient and accurate methods for non-negative matrix factorization problems applied to real-world, complex biomedical data. The goal is to help answer foremost biomedical questions of precision medicine: patient stratification, biomarker discovery and drugrepurposing.
The central mathematical optimization problem that we will study is penalized nonnegative matrix tri-factorization (PNMTF), which is a non-convex high-dimensional optimization problem, hence (unless P=NP) there is no efficient algorithm to solve it to optimality. Therefore, we will focus on developing state-of-the-art approximate algorithms using Fixed Point Method and Coordinate Descend Method combined with variants of first and second order methods. Analysis of theoretical and practical performance will be provided.
The central data science problem that we will work on is how to use the near optimum solutions of PNMTF to obtain good co-clustering and good multi-associations within the underlying heterogeneous networked data points. The bottleneck of this part is another NP-hard problem, the k-partite matching problem for k>=3. We will design new heuristics for k-partite matching (k>=3) carefully tuned to approximately solve this problem for our particular applications, and finally, develop state-of-the-art algorithms for co-clustering and multi-association detection of biomedical networked data.