Model-based Co-clustering for High Dimensional Sparse Data

Abstract We propose a novel model based on the von Mises-Fisher (vMF) distribution for coclustering high dimensional sparse matrices. While existing vMF-based models are only suitable for clustering along one dimension, our model acts simultaneously on both dimensions of a data matrix. Thereby it has the advantage of exploiting the inherent duality between rows and columns. Setting our model under the maximum likelihood (ML) approach and the classification ML (CML) approach, we derive two novel, hard and soft, co-clustering algorithms. Empirical results on numerous synthetic and real-world text datasets, demonstrate the effectiveness of our approach, for modelling high dimensional sparse data and co-clustering. Furthermore, thanks to our formulation, that performs an implicitly adaptive dimensionality reduction at each stage, our model alleviates the problem of high concentration parameters kappa's, a well known difficulty in the classical vMF-based models

Sparse machine learning models in bioinformatics

The meaning of parsimony is twofold in machine learning: either the structure or (and) the parameter of a model can be sparse. Sparse models have many strengths. First, sparsity is an important regularization principle to reduce model complexity and therefore avoid overfitting. Second, in many fields, for example bioinformatics, many high-dimensional data may be generated by a very few number of hidden factors, thus it is more reasonable to use a proper sparse model than a dense model. Third, a sparse model is often easy to interpret. In this dissertation, we investigate the sparse machine learning models and their applications in high-dimensional biological data analysis. We focus our research on five types of sparse models as follows. First, sparse representation is a parsimonious principle that a sample can be approximated by a sparse linear combination of basis vectors. We explore existing sparse representation models and propose our own sparse representation methods for high dimensional biological data analysis. We derive different sparse representation models from a Bayesian perspective. Two generic dictionary learning frameworks are proposed. Also, kernel and supervised dictionary learning approaches are devised. Furthermore, we propose fast active-set and decomposition methods for the optimization of sparse coding models. Second, gene-sample-time data are promising in clinical study, but challenging in computation. We propose sparse tensor decomposition methods and kernel methods for the dimensionality reduction and classification of such data. As the extensions of matrix factorization, tensor decomposition techniques can reduce the dimensionality of the gene-sample-time data dramatically, and the kernel methods can run very efficiently on such data. Third, we explore two sparse regularized linear models for multi-class problems in bioinformatics. Our first method is called the nearest-border classification technique for data with many classes. Our second method is a hierarchical model. It can simultaneously select features and classify samples. Our experiment, on breast tumor subtyping, shows that this model outperforms the one-versus-all strategy in some cases. Fourth, we propose to use spectral clustering approaches for clustering microarray time-series data. The approaches are based on two transformations that have been recently introduced, especially for gene expression time-series data, namely, alignment-based and variation-based transformations. Both transformations have been devised in order to take into account temporal relationships in the data, and have been shown to increase the ability of a clustering method in detecting co-expressed genes. We investigate the performances of these transformations methods, when combined with spectral clustering on two microarray time-series datasets, and discuss their strengths and weaknesses. Our experiments on two well known real-life datasets show the superiority of the alignment-based over the variation-based transformation for finding meaningful groups of co-expressed genes. Fifth, we propose the max-min high-order dynamic Bayesian network (MMHO-DBN) learning algorithm, in order to reconstruct time-delayed gene regulatory networks. Due to the small sample size of the training data and the power-low nature of gene regulatory networks, the structure of the network is restricted by sparsity. We also apply the qualitative probabilistic networks (QPNs) to interpret the interactions learned. Our experiments on both synthetic and real gene expression time-series data show that, MMHO-DBN can obtain better precision than some existing methods, and perform very fast. The QPN analysis can accurately predict types of influences and synergies. Additionally, since many high dimensional biological data are subject to missing values, we survey various strategies for learning models from incomplete data. We extend the existing imputation methods, originally for two-way data, to methods for gene-sample-time data. We also propose a pair-wise weighting method for computing kernel matrices from incomplete data. Computational evaluations show that both approaches work very robustly

Fast Robust PCA on Graphs

Mining useful clusters from high dimensional data has received significant attention of the computer vision and pattern recognition community in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix factorization methods) and susceptibility to gross corruptions in the data. In this paper we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient and scalable for huge datasets with O(nlog(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data

Discriminative variable selection for clustering with the sparse Fisher-EM algorithm

The interest in variable selection for clustering has increased recently due to the growing need in clustering high-dimensional data. Variable selection allows in particular to ease both the clustering and the interpretation of the results. Existing approaches have demonstrated the efficiency of variable selection for clustering but turn out to be either very time consuming or not sparse enough in high-dimensional spaces. This work proposes to perform a selection of the discriminative variables by introducing sparsity in the loading matrix of the Fisher-EM algorithm. This clustering method has been recently proposed for the simultaneous visualization and clustering of high-dimensional data. It is based on a latent mixture model which fits the data into a low-dimensional discriminative subspace. Three different approaches are proposed in this work to introduce sparsity in the orientation matrix of the discriminative subspace through $\ell_{1}$-type penalizations. Experimental comparisons with existing approaches on simulated and real-world data sets demonstrate the interest of the proposed methodology. An application to the segmentation of hyperspectral images of the planet Mars is also presented