Matrix co-factorization involves jointly decomposing several data matrices to approximate each data matrix as a product of two factor matrices, sharing some factor matrices in the factorization. We have recently developed variational Bayesian matrix co-factorization where factor matrices are inferred by computing variational posterior distributions in the case of Gaussian likelihood with Gaussian prior placed on factor matrices. Empirical Bayesian method was used, so hyperparameters are set to specific values determined by maximizing marginal likelihood. In this paper we present a hierarchical Bayesian model for matrix co-factorization in which we derive a variational inference algorithm to approximately compute posterior distributions over factor matrices as well as hyperparameters, placing Gaussian-Wishart prior on hyperparameters. Numerical experiments on MovieLens data demonstrate that the hierarchical variational Bayesian matrix co-factorization alleviates the over-fitting better than the empirical variational Bayesian matrix co-factorization, leading to the improved performance in terms of MAE and RMSE.

MOTIVATION: Systems biology attempts to describe complex systems behaviors in terms of dynamic operations of biological networks. However, there is lack of tools that can effectively decode complex network dynamics over multiple conditions. RESULTS: We present principal network analysis (PNA) that can automatically capture major dynamic activation patterns over multiple conditions and then generate protein and metabolic subnetworks for the captured patterns. We first demonstrated the utility of this method by applying it to a synthetic dataset. The results showed that PNA correctly captured the subnetworks representing dynamics in the data. We further applied PNA to two time-course gene expression profiles collected from (i) MCF7 cells after treatments of HRG at multiple doses and (ii) brain samples of four strains of mice infected with two prion strains. The resulting subnetworks and their interactions revealed network dynamics associated with HRG dose-dependent regulation of cell proliferation and differentiation and early PrPSc accumulation during prion infection. AVAILABILITY: The web-based software is available at: http://sbm.postech.ac.kr/pna.

An unsupervised method is proposed aiming at extracting rhythmic sources from commercial polyphonic music whose number of channels is limited to one. Commercial music signals are not usually provided with more than two channels while they often contain multiple instruments including singing voice. Therefore, instead of using conventional ways, such as modeling mixing environments or statistical characteristics, we should introduce other source-specific characteristics for separating or extracting the sources. In this paper, we concentrate on extracting rhythmic sources from the mixture with the other harmonic sources. An extension of nonnegative matrix factorization (NMF) is used to analyze multiple relationships between spectral and temporal properties in the given input matrices. Moreover, temporal repeatability of the rhythmic sound sources is implicated as common rhythmic property among segments of an input mixture signal. The proposed method shows acceptable, but not superior separation quality to the referred drum source separation systems. However, it has better applicability due to its blind manner in separation.

We address a problem of separating drums from polyphonic music containing various pitched instruments as well as drums. Nonnegative matrix factorization (NMF) was successfully applied to spectrograms of music to learn basis vectors, followed by support vector machine (SVM) to classify basis vectors into ones associated with drums (rhythmic source) only and pitched instruments (harmonic sources). Basis vectors associated with pitched instruments are used to reconstruct drum-eliminated music. However, it is cumbersome to construct a training set for pitched instruments since various instruments are involved. In this paper, we propose a method which only incorporates prior knowledge on drums, not requiring such training sets of pitched instruments. To this end, we present nonnegative matrix partial co-factorization (NMPCF) where the target matrix (spectrograms of music) and drum-only-matrix (collected from various drums a priori) are simultaneously decomposed, sharing some factor matrix partially, to force some portion of basis vectors to be associated with drums only. We develop a simple multiplicative algorithm for NMPCF and show its usefulness empirically, with numerical experiments on real-world music signals.

Nonnegative matrix factorization (NMF) is a popular method for low-rank approximation of nonnegative matrix, providing a useful tool for representation learning that is valuable for clustering and classification. When a portion of data are labeled, the performance of clustering or classification is improved if the information on class labels is incorporated into NMF. To this end, we present semi-supervised NMF (SSNMF), where we jointly incorporate the data matrix and the (partial) class label matrix into NMF. We develop multiplicative updates for SSNMF to minimize a sum of weighted residuals, each of which involves the nonnegative 2-factor decomposition of the data matrix or the label matrix, sharing a common factor matrix. Experiments on document datasets and EEG datasets in BCI competition confirm that our method improves clustering as well as classification performance, compared to the standard NMF, stressing that semi-supervised NMF yields semi-supervised feature extraction.

In this paper we present a method for extracting a speech signal of target speaker from noisy convolutive mixtures of target speech and an interference source, when training utterances of the target speaker are available. We incorporate a statistical latent variable model into blind source separation (BSS), where we make use of spectral bases learned from the training utterances of the target speaker to identify which source corresponds to the target speaker. Combined with any existing BSS methods, our post-processing (which is the main contribution) consists of two steps: (1) channel selection where we identify the source corresponding to the target speaker; (2) enhancement where we further suppress the remaining interference. Numerical experiments confirm that our method substantially improves the separation quality of existing BSS methods and successfully restores the target speaker's speech.

Nonnegative matrix tri-factorization (NMTF) is a 3-factor decomposition of a nonnegative data matrix, X ap USVT, where factor matrices, U, S, and V , are restricted to be nonnegative as well. Motivated by the aspect model used for dyadic data analysis as well as in probabilistic latent semantic analysis (PLSA), we present a probabilistic model with two dependent latent variables for NMTF, referred to as probabilistic matrix tri-factorization (PMTF). Each latent variable in the model is associated with the cluster variable for the corresponding object in the dyad, leading the model suited to co-clustering. We develop an EM algorithm to learn the PMTF model, showing its equivalence to multiplicative updates derived by an algebraic approach. We demonstrate the useful behavior of PMTF in a task of document clustering. Moreover, we incorporate the likelihood in the PMTF model into existing information criteria so that the number of clusters can be detected, while the algebraic NMTF cannot.