In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We present a family of fast recursive algorithms, and prove they are robust under any small perturbations of the input data matrix. This family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance.
Université de Mons（蒙斯大学）；University of Waterloo（滑铁卢大学）
Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing. This problem is referred to as near-separable NMF and requires that there exists a cone spanned by a small subset of the columns of the input nonnegative matrix approximately containing all columns. In this paper, we propose a preconditioning based on semidefinite programming making the input matrix well-conditioned. This in turn can improve significantly t…
University of Mons（蒙斯大学）；Univ Rennes（雷恩大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；IRISA（法国雷恩信息与自动化系统研究所）
We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns of the first NMF factor are equal to columns of the input matrix, while sparsity requires that the columns of the second NMF factor are sparse. We call this variant sparse separable NMF (SSNMF), which we prove to be NP-complete, as opposed to separable NMF which can be solved in polynomial time. The main motivation to consider this new model is to handle underdetermined blind source separation problems, such as multispectral image unmixing.…
Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We disc…
Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data with applications such as image analysis, text mining, audio source separation and hyperspectral unmixing. Given a data matrix $M$ and a factorization rank $r$, NMF looks for a nonnegative matrix $W$ with $r$ columns and a nonnegative matrix $H$ with $r$ rows such that $M \approx WH$. NMF is NP-hard to solve in general. However, it can be computed efficiently under the separability assumption which requires that the basis vectors appear as data points, that is, that there exists an index set $\math…
Scalable methods for nonnegative matrix factorizations of near-separable tall-and-skinny matrices
Stanford University（斯坦福大学）；Purdue University（普渡大学）；Purdue University Institute for Computational and Bindley Biosciences Center（普渡大学计算与Bindley生物科学中心）；Computer Science Mathematical Engineering（计算机科学数学工程）
Stanford University（斯坦福大学）；Université de Mons（蒙斯大学）
In this paper, we propose a provably correct algorithm for convolutive nonnegative matrix factorization (CNMF) under separability assumptions. CNMF is a convolutive variant of nonnegative matrix factorization (NMF), which functions as an NMF with additional sequential structure. This model is useful in a number of applications, such as audio source separation and neural sequence identification. While a number of heuristic algorithms have been proposed to solve CNMF, to the best of our knowledge no provably correct algorithms have been developed. We present an algorithm that takes advantage of …
Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences. To address this, in this work we propose to use structured random compression, that is, random projections that exploit the data structure, for two NMF variants: classical and separable. In separable NMF (SNMF) the left factors are a subset of the column…
Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this property of NMF on three applications, in image processing, text mining and hyperspectral imaging --this is the why. Then we address the problem of solving NMF, which is NP-hard in general. We review some standard NMF algorithms, and also present a recent subclass of NMF problems, referred to as near-separable NMF, that can be solved efficiently (that is, in pol…
Oregon State University（俄勒冈州立大学）；The Chinese University of Hong Kong（香港中文大学）
Nonnegative matrix factorization (NMF) often relies on the separability condition for tractable algorithm design. Separability-based NMF is mainly handled by two types of approaches, namely, greedy pursuit and convex programming. A notable convex NMF formulation is the so-called self-dictionary multiple measurement vectors (SD-MMV), which can work without knowing the matrix rank a priori, and is arguably more resilient to error propagation relative to greedy pursuit. However, convex SD-MMV renders a large memory cost that scales quadratically with the problem size. This memory challenge has be…
We introduce a new method based on nonnegative matrix factorization, Neural NMF, for detecting latent hierarchical structure in data. Datasets with hierarchical structure arise in a wide variety of fields, such as document classification, image processing, and bioinformatics. Neural NMF recursively applies NMF in layers to discover overarching topics encompassing the lower-level features. We derive a backpropagation optimization scheme that allows us to frame hierarchical NMF as a neural network. We test Neural NMF on a synthetic hierarchical dataset, the 20 Newsgroups dataset, and the MyLymeD…
In this paper, a new method is proposed for sparse PCA based on the recursive divide-and-conquer methodology. The main idea is to separate the original sparse PCA problem into a series of much simpler sub-problems, each having a closed-form solution. By recursively solving these sub-problems in an analytical way, an efficient algorithm is constructed to solve the sparse PCA problem. The algorithm only involves simple computations and is thus easy to implement. The proposed method can also be very easily extended to other sparse PCA problems with certain constraints, such as the nonnegative spa…
A Provably-Correct and Robust Convex Model for Smooth Separable NMF
Department of Mathematics, Hong Kong Baptist University, Hong Kong（香港 Baptist 大学数学系）；Institute of Data Science and Artificial Intelligence, Faculty of Computer Science and Engineering, Innopolis University（Innopolis 大学数据科学与人工智能研究院）；Department of Mathematics and Operational Research, Université de Mons, Rue de Houdain 9, 7000 Mons, Belgium（蒙斯大学数学与运筹系）；European Research Council (ERC consolidator, eLinoR, no 101085607)（欧洲研究理事会（ERC consolidator, eLinoR, no 101085607））
The successive projection algorithm (SPA) is a fast algorithm to tackle separable nonnegative matrix factorization (NMF). Given a nonnegative data matrix $X$, SPA identifies an index set $\mathcal{K}$ such that there exists a nonnegative matrix $H$ with $X \approx X(:,\mathcal{K})H$. SPA has been successfully used as a pure-pixel search algorithm in hyperspectral unmixing and for anchor word selection in document classification. Moreover, SPA is provably robust in low-noise settings. The main drawbacks of SPA are that it is not robust to outliers and does not take the data fitting term into ac…
Technische Universität Berlin（柏林工业大学）；Machine Learning Group（机器学习组）
While nonparametric density estimators often perform well on low dimensional data, their performance can suffer when applied to higher dimensional data, owing presumably to the curse of dimensionality. One technique for avoiding this is to assume no dependence between features and that the data are sampled from a separable density. This allows one to estimate each marginal distribution independently thereby avoiding the slow rates associated with estimating the full joint density. This is a strategy employed in naive Bayes models and is analogous to estimating a rank-one tensor. In this paper …
Sound and Image Processing Laboratory, School of Electrical Engineering, KTH Royal Institute of Technology（瑞典皇家理工学院电气工程学院声音与图像处理实验室）；Zenuity AB
Deriving a good model for multitalker babble noise can facilitate different speech processing algorithms, e.g. noise reduction, to reduce the so-called cocktail party difficulty. In the available systems, the fact that the babble waveform is generated as a sum of N different speech waveforms is not exploited explicitly. In this paper, first we develop a gamma hidden Markov model for power spectra of the speech signal, and then formulate it as a sparse nonnegative matrix factorization (NMF). Second, the sparse NMF is extended by relaxing the sparsity constraint, and a novel model for babble noi…
Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, Ré and Tropp (`Factoring nonnegative matrices with linear programs', NIPS 2012) proposed a linear programming (LP) model, referred to as Hottopixx, which is robust under any small perturbation of the input matrix. However, Hottopixx has two important drawbacks: (i) the input matrix has to be normalized, and (ii) the factorization rank has to be kn…
Information divergence that measures the difference between two nonnegative matrices or tensors has found its use in a variety of machine learning problems. Examples are Nonnegative Matrix/Tensor Factorization, Stochastic Neighbor Embedding, topic models, and Bayesian network optimization. The success of such a learning task depends heavily on a suitable divergence. A large variety of divergences have been suggested and analyzed, but very few results are available for an objective choice of the optimal divergence for a given task. Here we present a framework that facilitates automatic selectio…
The successive projection algorithm (SPA) has been known to work well for separable nonnegative matrix factorization (NMF) problems arising in applications, such as topic extraction from documents and endmember detection in hyperspectral images. One of the reasons is in that the algorithm is robust to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677-698, 2015] that a preconditioner can further enhance its noise robustness. The proof rested on the condition that the dimension $d$ and factorization rank $r$ in the separable NMF problem coincide with each other. However, it may…
Télécom Paris, Institut Polytechnique de Paris（巴黎电信学院，巴黎综合理工学院）；University of Mons（蒙斯大学）；University of Toulouse, IRIT/INP-ENSEEIHT（图卢兹大学，IRIT/INP-ENSEEIHT）
In this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme,…
We present DCSVM, an efficient algorithm for multi-class classification using Support Vector Machines. DCSVM is a divide and conquer algorithm which relies on data sparsity in high dimensional space and performs a smart partitioning of the whole training data set into disjoint subsets that are easily separable. A single prediction performed between two partitions eliminates at once one or more classes in one partition, leaving only a reduced number of candidate classes for subsequent steps. The algorithm continues recursively, reducing the number of classes at each step, until a final binary d…
