Codimensional matrix pairing perspective of BYY harmony learning: hierarchy of bilinear systems, joint decomposition of data-covariance, and applications of network biology

Abstract

One paper in a preceding issue of this journal has introduced the Bayesian Ying-Yang (BYY) harmony learning from a perspective of problem solving, parameter learning, and model selection. In a complementary role, the paper provides further insights from another perspective that a co-dimensional matrix pair (shortly co-dim matrix pair) forms a building unit and a hierarchy of such building units sets up the BYY system. The BYY harmony learning is re-examined via exploring the nature of a co-dim matrix pair, which leads to improved learning performance with refined model selection criteria and a modified mechanism that coordinates automatic model selection and sparse learning. Besides updating typical algorithms of factor analysis (FA), binary FA (BFA), binary matrix factorization (BMF), and nonnegative matrix factorization (NMF) to share such a mechanism, we are also led to (a) a new parametrization that embeds a de-noise nature to Gaussian mixture and local FA (LFA); (b) an alternative formulation of graph Laplacian based linear manifold learning; (c) a codecomposition of data and covariance for learning regularization and data integration; and (d) a co-dim matrix pair based generalization of temporal FA and state space model. Moreover, with help of a co-dim matrix pair in Hadamard product, we are led to a semi-supervised formation for regression analysis and a semi-blind learning formation for temporal FA and state space model. Furthermore, we address that these advances provide with new tools for network biology studies, including learning transcriptional regulatory, Protein-Protein Interaction network alignment, and network integration.