Robust Unsupervised Flexible Auto-weighted Local-Coordinate Concept Factorization for Image Clustering

We investigate the high-dimensional data clustering problem by proposing a novel and unsupervised representation learning model called Robust Flexible Auto-weighted Local-coordinate Concept Factorization (RFA-LCF). RFA-LCF integrates the robust flexible CF, robust sparse local-coordinate coding and the adaptive reconstruction weighting learning into a unified model. The adaptive weighting is driven by including the joint manifold preserving constraints on the recovered clean data, basis concepts and new representation. Specifically, our RFA-LCF uses a L2,1-norm based flexible residue to encode the mismatch between clean data and its reconstruction, and also applies the robust adaptive sparse local-coordinate coding to represent the data using a few nearby basis concepts, which can make the factorization more accurate and robust to noise. The robust flexible factorization is also performed in the recovered clean data space for enhancing representations. RFA-LCF also considers preserving the local manifold structures of clean data space, basis concept space and the new coordinate space jointly in an adaptive manner way. Extensive comparisons show that RFA-LCF can deliver enhanced clustering results.Comment: Accepted at the 44th IEEE International Conference on Acoustics, Speech, and Signal Processing(ICASSP 2019

Sparse Deep Nonnegative Matrix Factorization

Nonnegative matrix factorization is a powerful technique to realize dimension reduction and pattern recognition through single-layer data representation learning. Deep learning, however, with its carefully designed hierarchical structure, is able to combine hidden features to form more representative features for pattern recognition. In this paper, we proposed sparse deep nonnegative matrix factorization models to analyze complex data for more accurate classification and better feature interpretation. Such models are designed to learn localized features or generate more discriminative representations for samples in distinct classes by imposing $L_1$-norm penalty on the columns of certain factors. By extending one-layer model into multi-layer one with sparsity, we provided a hierarchical way to analyze big data and extract hidden features intuitively due to nonnegativity. We adopted the Nesterov's accelerated gradient algorithm to accelerate the computing process with the convergence rate of $O(1/k^2)$ after $k$ steps iteration. We also analyzed the computing complexity of our framework to demonstrate their efficiency. To improve the performance of dealing with linearly inseparable data, we also considered to incorporate popular nonlinear functions into this framework and explored their performance. We applied our models onto two benchmarking image datasets, demonstrating our models can achieve competitive or better classification performance and produce intuitive interpretations compared with the typical NMF and competing multi-layer models.Comment: 13 pages, 8 figure

Effective Spectral Unmixing via Robust Representation and Learning-based Sparsity

Hyperspectral unmixing (HU) plays a fundamental role in a wide range of hyperspectral applications. It is still challenging due to the common presence of outlier channels and the large solution space. To address the above two issues, we propose a novel model by emphasizing both robust representation and learning-based sparsity. Specifically, we apply the $\ell_{2,1}$-norm to measure the representation error, preventing outlier channels from dominating our objective. In this way, the side effects of outlier channels are greatly relieved. Besides, we observe that the mixed level of each pixel varies over image grids. Based on this observation, we exploit a learning-based sparsity method to simultaneously learn the HU results and a sparse guidance map. Via this guidance map, the sparsity constraint in the $\ell_{p}\!\left(\!0\!<\! p\!\leq\!1\right)$-norm is adaptively imposed according to the learnt mixed level of each pixel. Compared with state-of-the-art methods, our model is better suited to the real situation, thus expected to achieve better HU results. The resulted objective is highly non-convex and non-smooth, and so it is hard to optimize. As a profound theoretical contribution, we propose an efficient algorithm to solve it. Meanwhile, the convergence proof and the computational complexity analysis are systematically provided. Extensive evaluations verify that our method is highly promising for the HU task---it achieves very accurate guidance maps and much better HU results compared with state-of-the-art methods

Spectral Unmixing via Data-guided Sparsity

Hyperspectral unmixing, the process of estimating a common set of spectral bases and their corresponding composite percentages at each pixel, is an important task for hyperspectral analysis, visualization and understanding. From an unsupervised learning perspective, this problem is very challenging---both the spectral bases and their composite percentages are unknown, making the solution space too large. To reduce the solution space, many approaches have been proposed by exploiting various priors. In practice, these priors would easily lead to some unsuitable solution. This is because they are achieved by applying an identical strength of constraints to all the factors, which does not hold in practice. To overcome this limitation, we propose a novel sparsity based method by learning a data-guided map to describe the individual mixed level of each pixel. Through this data-guided map, the $\ell_{p}(0<p<1)$ constraint is applied in an adaptive manner. Such implementation not only meets the practical situation, but also guides the spectral bases toward the pixels under highly sparse constraint. What's more, an elegant optimization scheme as well as its convergence proof have been provided in this paper. Extensive experiments on several datasets also demonstrate that the data-guided map is feasible, and high quality unmixing results could be obtained by our method

A Unified Joint Matrix Factorization Framework for Data Integration

Nonnegative matrix factorization (NMF) is a powerful tool in data exploratory analysis by discovering the hidden features and part-based patterns from high-dimensional data. NMF and its variants have been successfully applied into diverse fields such as pattern recognition, signal processing, data mining, bioinformatics and so on. Recently, NMF has been extended to analyze multiple matrices simultaneously. However, a unified framework is still lacking. In this paper, we introduce a sparse multiple relationship data regularized joint matrix factorization (JMF) framework and two adapted prediction models for pattern recognition and data integration. Next, we present four update algorithms to solve this framework. The merits and demerits of these algorithms are systematically explored. Furthermore, extensive computational experiments using both synthetic data and real data demonstrate the effectiveness of JMF framework and related algorithms on pattern recognition and data mining.Comment: 14 pages, 7 figure

Accelerated Parallel and Distributed Algorithm using Limited Internal Memory for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a powerful technique for dimension reduction, extracting latent factors and learning part-based representation. For large datasets, NMF performance depends on some major issues: fast algorithms, fully parallel distributed feasibility and limited internal memory. This research aims to design a fast fully parallel and distributed algorithm using limited internal memory to reach high NMF performance for large datasets. In particular, we propose a flexible accelerated algorithm for NMF with all its $L_1$ $L_2$ regularized variants based on full decomposition, which is a combination of an anti-lopsided algorithm and a fast block coordinate descent algorithm. The proposed algorithm takes advantages of both these algorithms to achieve a linear convergence rate of $\mathcal{O}(1-\frac{1}{||Q||_2})^k$ in optimizing each factor matrix when fixing the other factor one in the sub-space of passive variables, where $r$ is the number of latent components; where $\sqrt{r} \leq ||Q||_2 \leq r$. In addition, the algorithm can exploit the data sparseness to run on large datasets with limited internal memory of machines. Furthermore, our experimental results are highly competitive with 7 state-of-the-art methods about three significant aspects of convergence, optimality and average of the iteration number. Therefore, the proposed algorithm is superior to fast block coordinate descent methods and accelerated methods

A Nonlinear Orthogonal Non-Negative Matrix Factorization Approach to Subspace Clustering

A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal multiplicative update rules to solve the subspace clustering problem. In nonlinear orthogonal NMF framework, we propose two subspace clustering algorithms, named kernel-based non-negative subspace clustering KNSC-Ncut and KNSC-Rcut and establish their connection with spectral normalized cut and ratio cut clustering. We further extend the nonlinear orthogonal NMF framework and introduce a graph regularization to obtain a factorization that respects a local geometric structure of the data after the nonlinear mapping. The proposed NMF-based approach to subspace clustering takes into account the nonlinear nature of the manifold, as well as its intrinsic local geometry, which considerably improves the clustering performance when compared to the several recently proposed state-of-the-art methods

Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297, arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805, arXiv:1403.8067 by other authors, Computer Science Review, November 201

A Survey on Multi-View Clustering

With advances in information acquisition technologies, multi-view data become ubiquitous. Multi-view learning has thus become more and more popular in machine learning and data mining fields. Multi-view unsupervised or semi-supervised learning, such as co-training, co-regularization has gained considerable attention. Although recently, multi-view clustering (MVC) methods have been developed rapidly, there has not been a survey to summarize and analyze the current progress. Therefore, this paper reviews the common strategies for combining multiple views of data and based on this summary we propose a novel taxonomy of the MVC approaches. We further discuss the relationships between MVC and multi-view representation, ensemble clustering, multi-task clustering, multi-view supervised and semi-supervised learning. Several representative real-world applications are elaborated. To promote future development of MVC, we envision several open problems that may require further investigation and thorough examination.Comment: 17 pages, 4 figure

Beyond Alternating Updates for Matrix Factorization with Inertial Bregman Proximal Gradient Algorithms

Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization variables. A remedy is non-alternating schemes. However, due to a lack of Lipschitz continuity of the gradient in matrix factorization problems, convergence cannot be guaranteed. A recently developed approach relies on the concept of Bregman distances, which generalizes the standard Euclidean distance. We exploit this theory by proposing a novel Bregman distance for matrix factorization problems, which, at the same time, allows for simple/closed form update steps. Therefore, for non-alternating schemes, such as the recently introduced Bregman Proximal Gradient (BPG) method and an inertial variant Convex--Concave Inertial BPG (CoCaIn BPG), convergence of the whole sequence to a stationary point is proved for Matrix Factorization. In several experiments, we observe a superior performance of our non-alternating schemes in terms of speed and objective value at the limit point.Comment: Accepted at NeuRIPS 2019. Paper url: http://papers.nips.cc/paper/8679-beyond-alternating-updates-for-matrix-factorization-with-inertial-bregman-proximal-gradient-algorithm