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

Learning From Hidden Traits: Joint Factor Analysis and Latent Clustering

Dimensionality reduction techniques play an essential role in data analytics, signal processing and machine learning. Dimensionality reduction is usually performed in a preprocessing stage that is separate from subsequent data analysis, such as clustering or classification. Finding reduced-dimension representations that are well-suited for the intended task is more appealing. This paper proposes a joint factor analysis and latent clustering framework, which aims at learning cluster-aware low-dimensional representations of matrix and tensor data. The proposed approach leverages matrix and tensor factorization models that produce essentially unique latent representations of the data to unravel latent cluster structure -- which is otherwise obscured because of the freedom to apply an oblique transformation in latent space. At the same time, latent cluster structure is used as prior information to enhance the performance of factorization. Specific contributions include several custom-built problem formulations, corresponding algorithms, and discussion of associated convergence properties. Besides extensive simulations, real-world datasets such as Reuters document data and MNIST image data are also employed to showcase the effectiveness of the proposed approaches

Global and Local Structure Preserving Sparse Subspace Learning: An Iterative Approach to Unsupervised Feature Selection

As we aim at alleviating the curse of high-dimensionality, subspace learning is becoming more popular. Existing approaches use either information about global or local structure of the data, and few studies simultaneously focus on global and local structures as the both of them contain important information. In this paper, we propose a global and local structure preserving sparse subspace learning (GLoSS) model for unsupervised feature selection. The model can simultaneously realize feature selection and subspace learning. In addition, we develop a greedy algorithm to establish a generic combinatorial model, and an iterative strategy based on an accelerated block coordinate descent is used to solve the GLoSS problem. We also provide whole iterate sequence convergence analysis of the proposed iterative algorithm. Extensive experiments are conducted on real-world datasets to show the superiority of the proposed approach over several state-of-the-art unsupervised feature selection approaches.Comment: 32 page, 6 figures and 60 reference

Supervised Nonnegative Matrix Factorization to Predict ICU Mortality Risk

ICU mortality risk prediction is a tough yet important task. On one hand, due to the complex temporal data collected, it is difficult to identify the effective features and interpret them easily; on the other hand, good prediction can help clinicians take timely actions to prevent the mortality. These correspond to the interpretability and accuracy problems. Most existing methods lack of the interpretability, but recently Subgraph Augmented Nonnegative Matrix Factorization (SANMF) has been successfully applied to time series data to provide a path to interpret the features well. Therefore, we adopted this approach as the backbone to analyze the patient data. One limitation of the raw SANMF method is its poor prediction ability due to its unsupervised nature. To deal with this problem, we proposed a supervised SANMF algorithm by integrating the logistic regression loss function into the NMF framework and solved it with an alternating optimization procedure. We used the simulation data to verify the effectiveness of this method, and then we applied it to ICU mortality risk prediction and demonstrated its superiority over other conventional supervised NMF methods.Comment: 7 Pages, 2 figure

Long-Term Identity-Aware Multi-Person Tracking for Surveillance Video Summarization

Multi-person tracking plays a critical role in the analysis of surveillance video. However, most existing work focus on shorter-term (e.g. minute-long or hour-long) video sequences. Therefore, we propose a multi-person tracking algorithm for very long-term (e.g. month-long) multi-camera surveillance scenarios. Long-term tracking is challenging because 1) the apparel/appearance of the same person will vary greatly over multiple days and 2) a person will leave and re-enter the scene numerous times. To tackle these challenges, we leverage face recognition information, which is robust to apparel change, to automatically reinitialize our tracker over multiple days of recordings. Unfortunately, recognized faces are unavailable oftentimes. Therefore, our tracker propagates identity information to frames without recognized faces by uncovering the appearance and spatial manifold formed by person detections. We tested our algorithm on a 23-day 15-camera data set (4,935 hours total), and we were able to localize a person 53.2% of the time with 69.8% precision. We further performed video summarization experiments based on our tracking output. Results on 116.25 hours of video showed that we were able to generate a reasonable visual diary (i.e. a summary of what a person did) for different people, thus potentially opening the door to automatic summarization of the vast amount of surveillance video generated every day

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

Group Component Analysis for Multiblock Data: Common and Individual Feature Extraction

Very often data we encounter in practice is a collection of matrices rather than a single matrix. These multi-block data are naturally linked and hence often share some common features and at the same time they have their own individual features, due to the background in which they are measured and collected. In this study we proposed a new scheme of common and individual feature analysis (CIFA) that processes multi-block data in a linked way aiming at discovering and separating their common and individual features. According to whether the number of common features is given or not, two efficient algorithms were proposed to extract the common basis which is shared by all data. Then feature extraction is performed on the common and the individual spaces separately by incorporating the techniques such as dimensionality reduction and blind source separation. We also discussed how the proposed CIFA can significantly improve the performance of classification and clustering tasks by exploiting common and individual features of samples respectively. Our experimental results show some encouraging features of the proposed methods in comparison to the state-of-the-art methods on synthetic and real data.Comment: 13 pages,11 figure

Supervised Dictionary Learning and Sparse Representation-A Review

Dictionary learning and sparse representation (DLSR) is a recent and successful mathematical model for data representation that achieves state-of-the-art performance in various fields such as pattern recognition, machine learning, computer vision, and medical imaging. The original formulation for DLSR is based on the minimization of the reconstruction error between the original signal and its sparse representation in the space of the learned dictionary. Although this formulation is optimal for solving problems such as denoising, inpainting, and coding, it may not lead to optimal solution in classification tasks, where the ultimate goal is to make the learned dictionary and corresponding sparse representation as discriminative as possible. This motivated the emergence of a new category of techniques, which is appropriately called supervised dictionary learning and sparse representation (S-DLSR), leading to more optimal dictionary and sparse representation in classification tasks. Despite many research efforts for S-DLSR, the literature lacks a comprehensive view of these techniques, their connections, advantages and shortcomings. In this paper, we address this gap and provide a review of the recently proposed algorithms for S-DLSR. We first present a taxonomy of these algorithms into six categories based on the approach taken to include label information into the learning of the dictionary and/or sparse representation. For each category, we draw connections between the algorithms in this category and present a unified framework for them. We then provide guidelines for applied researchers on how to represent and learn the building blocks of an S-DLSR solution based on the problem at hand. This review provides a broad, yet deep, view of the state-of-the-art methods for S-DLSR and allows for the advancement of research and development in this emerging area of research

Robust Multi-subspace Analysis Using Novel Column L0-norm Constrained Matrix Factorization

We study the underlying structure of data (approximately) generated from a union of independent subspaces. Traditional methods learn only one subspace, failing to discover the multi-subspace structure, while state-of-the-art methods analyze the multi-subspace structure using data themselves as the dictionary, which cannot offer the explicit basis to span each subspace and are sensitive to errors via an indirect representation. Additionally, they also suffer from a high computational complexity, being quadratic or cubic to the sample size. To tackle all these problems, we propose a method, called Matrix Factorization with Column L0-norm constraint (MFC0), that can simultaneously learn the basis for each subspace, generate a direct sparse representation for each data sample, as well as removing errors in the data in an efficient way. Furthermore, we develop a first-order alternating direction algorithm, whose computational complexity is linear to the sample size, to stably and effectively solve the nonconvex objective function and non- smooth l0-norm constraint of MFC0. Experimental results on both synthetic and real-world datasets demonstrate that besides the superiority over traditional and state-of-the-art methods for subspace clustering, data reconstruction, error correction, MFC0 also shows its uniqueness for multi-subspace basis learning and direct sparse representation.Comment: 13 pages, 8 figures, 8 table

Feature Weighted Non-negative Matrix Factorization

Non-negative Matrix Factorization (NMF) is one of the most popular techniques for data representation and clustering, and has been widely used in machine learning and data analysis. NMF concentrates the features of each sample into a vector, and approximates it by the linear combination of basis vectors, such that the low-dimensional representations are achieved. However, in real-world applications, the features are usually with different importances. To exploit the discriminative features, some methods project the samples into the subspace with a transformation matrix, which disturbs the original feature attributes and neglects the diversity of samples. To alleviate the above problems, we propose the Feature weighted Non-negative Matrix Factorization (FNMF) in this paper. The salient properties of FNMF can be summarized as threefold: 1) it learns the weights of features adaptively according to their importances; 2) it utilizes multiple feature weighting components to preserve the diversity; 3) it can be solved efficiently with the suggested optimization algorithm. Performance on synthetic and real-world datasets demonstrate that the proposed method obtains the state-of-the-art performance