Exactly Robust Kernel Principal Component Analysis

Robust principal component analysis (RPCA) can recover low-rank matrices when they are corrupted by sparse noises. In practice, many matrices are, however, of high-rank and hence cannot be recovered by RPCA. We propose a novel method called robust kernel principal component analysis (RKPCA) to decompose a partially corrupted matrix as a sparse matrix plus a high or full-rank matrix with low latent dimensionality. RKPCA can be applied to many problems such as noise removal and subspace clustering and is still the only unsupervised nonlinear method robust to sparse noises. Our theoretical analysis shows that, with high probability, RKPCA can provide high recovery accuracy. The optimization of RKPCA involves nonconvex and indifferentiable problems. We propose two nonconvex optimization algorithms for RKPCA. They are alternating direction method of multipliers with backtracking line search and proximal linearized minimization with adaptive step size. Comparative studies in noise removal and robust subspace clustering corroborate the effectiveness and superiority of RKPCA.Comment: The paper was accepted by IEEE Transactions on Neural Networks and Learning System

Kernelized Low Rank Representation on Grassmann Manifolds

Low rank representation (LRR) has recently attracted great interest due to its pleasing efficacy in exploring low-dimensional subspace structures embedded in data. One of its successful applications is subspace clustering which means data are clustered according to the subspaces they belong to. In this paper, at a higher level, we intend to cluster subspaces into classes of subspaces. This is naturally described as a clustering problem on Grassmann manifold. The novelty of this paper is to generalize LRR on Euclidean space onto an LRR model on Grassmann manifold in a uniform kernelized framework. The new methods have many applications in computer vision tasks. Several clustering experiments are conducted on handwritten digit images, dynamic textures, human face clips and traffic scene sequences. The experimental results show that the proposed methods outperform a number of state-of-the-art subspace clustering methods.Comment: 13 page

Low Rank Representation on Grassmann Manifolds: An Extrinsic Perspective

Many computer vision algorithms employ subspace models to represent data. The Low-rank representation (LRR) has been successfully applied in subspace clustering for which data are clustered according to their subspace structures. The possibility of extending LRR on Grassmann manifold is explored in this paper. Rather than directly embedding Grassmann manifold into a symmetric matrix space, an extrinsic view is taken by building the self-representation of LRR over the tangent space of each Grassmannian point. A new algorithm for solving the proposed Grassmannian LRR model is designed and implemented. Several clustering experiments are conducted on handwritten digits dataset, dynamic texture video clips and YouTube celebrity face video data. The experimental results show our method outperforms a number of existing methods.Comment: 9 page

Kernelized LRR on Grassmann Manifolds for Subspace Clustering

Low rank representation (LRR) has recently attracted great interest due to its pleasing efficacy in exploring low-dimensional sub- space structures embedded in data. One of its successful applications is subspace clustering, by which data are clustered according to the subspaces they belong to. In this paper, at a higher level, we intend to cluster subspaces into classes of subspaces. This is naturally described as a clustering problem on Grassmann manifold. The novelty of this paper is to generalize LRR on Euclidean space onto an LRR model on Grassmann manifold in a uniform kernelized LRR framework. The new method has many applications in data analysis in computer vision tasks. The proposed models have been evaluated on a number of practical data analysis applications. The experimental results show that the proposed models outperform a number of state-of-the-art subspace clustering methods

Co-manifold learning with missing data

Representation learning is typically applied to only one mode of a data matrix, either its rows or columns. Yet in many applications, there is an underlying geometry to both the rows and the columns. We propose utilizing this coupled structure to perform co-manifold learning: uncovering the underlying geometry of both the rows and the columns of a given matrix, where we focus on a missing data setting. Our unsupervised approach consists of three components. We first solve a family of optimization problems to estimate a complete matrix at multiple scales of smoothness. We then use this collection of smooth matrix estimates to compute pairwise distances on the rows and columns based on a new multi-scale metric that implicitly introduces a coupling between the rows and the columns. Finally, we construct row and column representations from these multi-scale metrics. We demonstrate that our approach outperforms competing methods in both data visualization and clustering.Comment: 16 pages, 9 figure

Clustering is semidefinitely not that hard: Nonnegative SDP for manifold disentangling

In solving hard computational problems, semidefinite program (SDP) relaxations often play an important role because they come with a guarantee of optimality. Here, we focus on a popular semidefinite relaxation of K-means clustering which yields the same solution as the non-convex original formulation for well segregated datasets. We report an unexpected finding: when data contains (greater than zero-dimensional) manifolds, the SDP solution captures such geometrical structures. Unlike traditional manifold embedding techniques, our approach does not rely on manually defining a kernel but rather enforces locality via a nonnegativity constraint. We thus call our approach NOnnegative MAnifold Disentangling, or NOMAD. To build an intuitive understanding of its manifold learning capabilities, we develop a theoretical analysis of NOMAD on idealized datasets. While NOMAD is convex and the globally optimal solution can be found by generic SDP solvers with polynomial time complexity, they are too slow for modern datasets. To address this problem, we analyze a non-convex heuristic and present a new, convex and yet efficient, algorithm, based on the conditional gradient method. Our results render NOMAD a versatile, understandable, and powerful tool for manifold learning

Localized LRR on Grassmann Manifolds: An Extrinsic View

Subspace data representation has recently become a common practice in many computer vision tasks. It demands generalizing classical machine learning algorithms for subspace data. Low-Rank Representation (LRR) is one of the most successful models for clustering vectorial data according to their subspace structures. This paper explores the possibility of extending LRR for subspace data on Grassmann manifolds. Rather than directly embedding the Grassmann manifolds into the symmetric matrix space, an extrinsic view is taken to build the LRR self-representation in the local area of the tangent space at each Grassmannian point, resulting in a localized LRR method on Grassmann manifolds. A novel algorithm for solving the proposed model is investigated and implemented. The performance of the new clustering algorithm is assessed through experiments on several real-world datasets including MNIST handwritten digits, ballet video clips, SKIG action clips, DynTex++ dataset and highway traffic video clips. The experimental results show the new method outperforms a number of state-of-the-art clustering methodsComment: IEEE Transactions on Circuits and Systems for Video Technology with Minor Revisions. arXiv admin note: text overlap with arXiv:1504.0180

Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality

We consider the problem of decomposing a higher-order tensor with binary entries. Such data problems arise frequently in applications such as neuroimaging, recommendation system, topic modeling, and sensor network localization. We propose a multilinear Bernoulli model, develop a rank-constrained likelihood-based estimation method, and obtain the theoretical accuracy guarantees. In contrast to continuous-valued problems, the binary tensor problem exhibits an interesting phase transition phenomenon according to the signal-to-noise ratio. The error bound for the parameter tensor estimation is established, and we show that the obtained rate is minimax optimal under the considered model. Furthermore, we develop an alternating optimization algorithm with convergence guarantees. The efficacy of our approach is demonstrated through both simulations and analyses of multiple data sets on the tasks of tensor completion and clustering.Comment: 35 pages, 7 figures, 4 table

Group Preserving Label Embedding for Multi-Label Classification

Multi-label learning is concerned with the classification of data with multiple class labels. This is in contrast to the traditional classification problem where every data instance has a single label. Due to the exponential size of output space, exploiting intrinsic information in feature and label spaces has been the major thrust of research in recent years and use of parametrization and embedding have been the prime focus. Researchers have studied several aspects of embedding which include label embedding, input embedding, dimensionality reduction and feature selection. These approaches differ from one another in their capability to capture other intrinsic properties such as label correlation, local invariance etc. We assume here that the input data form groups and as a result, the label matrix exhibits a sparsity pattern and hence the labels corresponding to objects in the same group have similar sparsity. In this paper, we study the embedding of labels together with the group information with an objective to build an efficient multi-label classification. We assume the existence of a low-dimensional space onto which the feature vectors and label vectors can be embedded. In order to achieve this, we address three sub-problems namely; (1) Identification of groups of labels; (2) Embedding of label vectors to a low rank-space so that the sparsity characteristic of individual groups remains invariant; and (3) Determining a linear mapping that embeds the feature vectors onto the same set of points, as in stage 2, in the low-dimensional space. We compare our method with seven well-known algorithms on twelve benchmark data sets. Our experimental analysis manifests the superiority of our proposed method over state-of-art algorithms for multi-label learning

Low-Rank Modeling and Its Applications in Image Analysis

Low-rank modeling generally refers to a class of methods that solve problems by representing variables of interest as low-rank matrices. It has achieved great success in various fields including computer vision, data mining, signal processing and bioinformatics. Recently, much progress has been made in theories, algorithms and applications of low-rank modeling, such as exact low-rank matrix recovery via convex programming and matrix completion applied to collaborative filtering. These advances have brought more and more attentions to this topic. In this paper, we review the recent advance of low-rank modeling, the state-of-the-art algorithms, and related applications in image analysis. We first give an overview to the concept of low-rank modeling and challenging problems in this area. Then, we summarize the models and algorithms for low-rank matrix recovery and illustrate their advantages and limitations with numerical experiments. Next, we introduce a few applications of low-rank modeling in the context of image analysis. Finally, we conclude this paper with some discussions.Comment: To appear in ACM Computing Survey