Data Clustering And Visualization Through Matrix Factorization

Clustering is traditionally an unsupervised task which is to find natural groupings or clusters in multidimensional data based on perceived similarities among the patterns. The purpose of clustering is to extract useful information from unlabeled data. In order to present the extracted useful knowledge obtained by clustering in a meaningful way, data visualization becomes a popular and growing area of research field. Visualization can provide a qualitative overview of large and complex data sets, which help us the desired insight in truly understanding the phenomena of interest in data. The contribution of this dissertation is two-fold: Semi-Supervised Non-negative Matrix Factorization (SS-NMF) for data clustering/co-clustering and Exemplar-based data Visualization (EV) through matrix factorization. Compared to traditional data mining models, matrix-based methods are fast, easy to understand and implement, especially suitable to solve large-scale challenging problems in text mining, image grouping, medical diagnosis, and bioinformatics. In this dissertation, we present two effective matrix-based solutions in the new directions of data clustering and visualization. First, in many practical learning domains, there is a large supply of unlabeled data but limited labeled data, and in most cases it might be expensive to generate large amounts of labeled data. Traditional clustering algorithms completely ignore these valuable labeled data and thus are inapplicable to these problems. Consequently, semi-supervised clustering, which can incorporate the domain knowledge to guide a clustering algorithm, has become a topic of significant recent interest. Thus, we develop a Non-negative Matrix Factorization (NMF) based framework to incorporate prior knowledge into data clustering. Moreover, with the fast growth of Internet and computational technologies in the past decade, many data mining applications have advanced swiftly from the simple clustering of one data type to the co-clustering of multiple data types, usually involving high heterogeneity. To this end, we extend SS-NMF to perform heterogeneous data co-clustering. From a theoretical perspective, SS-NMF for data clustering/co-clustering is mathematically rigorous. The convergence and correctness of our algorithms are proved. In addition, we discuss the relationship between SS-NMF with other well-known clustering and co-clustering models. Second, most of current clustering models only provide the centroids (e.g., mathematical means of the clusters) without inferring the representative exemplars from real data, thus they are unable to better summarize or visualize the raw data. A new method, Exemplar-based Visualization (EV), is proposed to cluster and visualize an extremely large-scale data. Capitalizing on recent advances in matrix approximation and factorization, EV provides a means to visualize large scale data with high accuracy (in retaining neighbor relations), high efficiency (in computation), and high flexibility (through the use of exemplars). Empirically, we demonstrate the superior performance of our matrix-based data clustering and visualization models through extensive experiments performed on the publicly available large scale data sets

Sample Complexity of Dictionary Learning and other Matrix Factorizations

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), $K$-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to $\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix factorization techniques.Comment: to appea