We present a new machine learning framework called “self-taught learning ” for using unlabeled data in supervised classification tasks. We do not assume that the unlabeled data follows the same class labels or generative distribution as the labeled data. Thus, we would like to use a large number of unlabeled images (or audio samples, or text documents) randomly downloaded from the Internet to improve performance on a given image (or audio, or text) classification task. Such unlabeled data is significantly easier to obtain than in typical semi-supervised or transfer learning settings, making selftaught learning widely applicable to many practical learning problems. We describe an approach to self-taught learning that uses sparse coding to construct higher-level features using the unlabeled data. These features form a succinct input representation and significantly improve classification performance. When using an SVM for classification, we further show how a Fisher kernel can be learned for this representation. 1.

A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical

In this paper we discuss the development and use of low-rank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for large-scale and time-varying datasets. Key words: nonnegative matrix factorization, text mining, spectral data analysis, email surveillance, conjugate gradient, constrained least squares.

Currently, most research on nonnegative matrix factorization (NMF) focus on 2-factor X = FG T factorization. We provide a systematic analysis of 3-factor X = FSG T NMF. While unconstrained 3-factor NMF is equivalent to unconstrained 2-factor NMF, constrained 3factor NMF brings new features to constrained 2-factor NMF. We study the orthogonality constraint because it leads to rigorous clustering interpretation. We provide new rules for updating F,S,G and prove the convergence of these algorithms. Experiments on 5 datasets and a real world case study are performed to show the capability of bi-orthogonal 3-factor NMF on simultaneously clustering rows and columns of the input data matrix. We provide a new approach of evaluating the quality of clustering on words using class aggregate distribution and multi-peak distribution. We also provide an overview of various NMF extensions and examine their relationships.

Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing, non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed and optimization for both small and large datasets.

In this paper we discus a wide class of loss (cost) functions for non-negative matrix factorization (NMF) and derive several novel algorithms with improved efficiency and robustness to noise and outliers. We review several approaches which allow us to obtain generalized forms of multiplicative NMF algorithms and unify some existing algorithms. We give also the flexible and relaxed form of the NMF algorithms to increase convergence speed and impose some desired constraints such as sparsity and smoothness of components. Moreover, the effects of various regularization terms and constraints are clearly shown. The scope of these results is vast since the proposed generalized divergence functions include quite large number of useful loss functions such as the squared Euclidean distance,Kulback-Leibler divergence, Itakura-Saito, Hellinger, Pearson’s chi-square, and Neyman’s chi-square distances, etc. We have applied successfully the developed algorithms to blind (or semi blind) source separation (BSS) where sources can be generally statistically dependent, however they satisfy some other conditions or additional constraints such as nonnegativity, sparsity and/or smoothness.

We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F G T, we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods. 1

Abstract—We propose a novel nonnegative matrix factorization model that aims at finding localized, part-based, representations of nonnegative multivariate data items. Unlike the classical nonnegative matrix factorization (NMF) technique, this new model, denoted “nonsmooth nonnegative matrix factorization ” (nsNMF), corresponds to the optimization of an unambiguous cost function designed to explicitly represent sparseness, in the form of nonsmoothness, which is controlled by a single parameter. In general, this method produces a set of basis and encoding vectors that are not only capable of representing the original data, but they also extract highly localized patterns, which generally lend themselves to improved interpretability. The properties of this new method are illustrated with several data sets. Comparisons to previously published methods show that the new nsNMF method has some advantages in keeping faithfulness to the data in the achieving a high degree of sparseness for both the estimated basis and the encoding vectors and in better interpretability of the factors. Index Terms—nonnegative matrix factorization, constrained optimization, datamining, mining methods and algorithms, pattern analysis, feature extraction or construction, sparse, structured, and very large systems. æ 1

Abstract. Recovering the pose of a person from single images is a challenging problem. This paper discusses a bottom-up approach that uses local image features to estimate human upper body pose from single images in cluttered backgrounds. The method takes the image window with a dense grid of local gradient orientation histograms, followed by non negative matrix factorization to learn a set of bases that correspond to local features on the human body, enabling selective encoding of human-like features in the presence of background clutter. Pose is then recovered by direct regression. This approach allows us to key on gradient patterns such as shoulder contours and bent elbows that are characteristic of humans and carry important pose information, unlike current regressive methods that either use weak limb detectors or require prior segmentation to work. The system is trained on a database of images with labelled poses. We show that it estimates pose with similar performance levels to current example-based methods, but unlike them it works in the presence of natural backgrounds, without any prior segmentation. 1.

The nonnegative matrix factorization (NMF) has been shown recently to be useful for clustering. Various extensions of NMF have also been proposed. In this paper we present an overview and theoretically analyze the relationships among them. In addition, we clarify previously unaddressed issues, such as NMF normalization, cluster posterior probabilty, and NMF algoritm convergence rate. Experiments are also conducted to empirically evaluate and compare various factorization methods.