Evolutionary Self-Expressive Models for Subspace Clustering

The problem of organizing data that evolves over time into clusters is encountered in a number of practical settings. We introduce evolutionary subspace clustering, a method whose objective is to cluster a collection of evolving data points that lie on a union of low-dimensional evolving subspaces. To learn the parsimonious representation of the data points at each time step, we propose a non-convex optimization framework that exploits the self-expressiveness property of the evolving data while taking into account representation from the preceding time step. To find an approximate solution to the aforementioned non-convex optimization problem, we develop a scheme based on alternating minimization that both learns the parsimonious representation as well as adaptively tunes and infers a smoothing parameter reflective of the rate of data evolution. The latter addresses a fundamental challenge in evolutionary clustering -- determining if and to what extent one should consider previous clustering solutions when analyzing an evolving data collection. Our experiments on both synthetic and real-world datasets demonstrate that the proposed framework outperforms state-of-the-art static subspace clustering algorithms and existing evolutionary clustering schemes in terms of both accuracy and running time, in a range of scenarios

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

Robust Non-Linear Matrix Factorization for Dictionary Learning, Denoising, and Clustering

Low dimensional nonlinear structure abounds in datasets across computer vision and machine learning. Kernelized matrix factorization techniques have recently been proposed to learn these nonlinear structures for denoising, classification, dictionary learning, and missing data imputation, by observing that the image of the matrix in a sufficiently large feature space is low-rank. However, these nonlinear methods fail in the presence of sparse noise or outliers. In this work, we propose a new robust nonlinear factorization method called Robust Non-Linear Matrix Factorization (RNLMF). RNLMF constructs a dictionary for the data space by factoring a kernelized feature space; a noisy matrix can then be decomposed as the sum of a sparse noise matrix and a clean data matrix that lies in a low dimensional nonlinear manifold. RNLMF is robust to sparse noise and outliers and scales to matrices with thousands of rows and columns. Empirically, RNLMF achieves noticeable improvements over baseline methods in denoising and clustering

Subspace Clustering with Missing and Corrupted Data

Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a weighted combination of the other samples, with weights of minimal $\ell_1$ norm, and then uses those learned weights to cluster the samples. SSC is stable in settings where each sample is contaminated by a relatively small amount of noise. However, when there is a significant amount of additive noise, or a considerable number of entries are missing, theoretical guarantees are scarce. In this paper, we study a robust variant of SSC and establish clustering guarantees in the presence of corrupted or missing data. We give explicit bounds on amount of noise and missing data that the algorithm can tolerate, both in deterministic settings and in a random generative model. Notably, our approach provides guarantees for higher tolerance to noise and missing data than existing analyses for this method. By design, the results hold even when we do not know the locations of the missing data; e.g., as in presence-only data.Comment: 31 pages, 2 figure

Neither Global Nor Local: A Hierarchical Robust Subspace Clustering For Image Data

In this paper, we consider the problem of subspace clustering in presence of contiguous noise, occlusion and disguise. We argue that self-expressive representation of data in current state-of-the-art approaches is severely sensitive to occlusions and complex real-world noises. To alleviate this problem, we propose a hierarchical framework that brings robustness of local patches-based representations and discriminant property of global representations together. This approach consists of 1) a top-down stage, in which the input data is subject to repeated division to smaller patches and 2) a bottom-up stage, in which the low rank embedding of local patches in field of view of a corresponding patch in upper level are merged on a Grassmann manifold. This summarized information provides two key information for the corresponding patch on the upper level: cannot-links and recommended-links. This information is employed for computing a self-expressive representation of each patch at upper levels using a weighted sparse group lasso optimization problem. Numerical results on several real data sets confirm the efficiency of our approach

Theoretical Analysis of Sparse Subspace Clustering with Missing Entries

Sparse Subspace Clustering (SSC) is a popular unsupervised machine learning method for clustering data lying close to an unknown union of low-dimensional linear subspaces; a problem with numerous applications in pattern recognition and computer vision. Even though the behavior of SSC for complete data is by now well-understood, little is known about its theoretical properties when applied to data with missing entries. In this paper we give theoretical guarantees for SSC with incomplete data, and analytically establish that projecting the zero-filled data onto the observation pattern of the point being expressed leads to a substantial improvement in performance. The main insight that stems from our analysis is that even though the projection induces additional missing entries, this is counterbalanced by the fact that the projected and zero-filled data are in effect incomplete points associated with the union of the corresponding projected subspaces, with respect to which the point being expressed is complete. The significance of this phenomenon potentially extends to the entire class of self-expressive methods

Low-Rank Representation over the Manifold of Curves

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function. In this paper we propose a method to analyse subspace structure of the functional data by using the state of the art Low-Rank Representation (LRR). Experimental evaluation on synthetic and real data reveals that this method massively outperforms conventional LRR in tasks concerning functional data

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

Cluster Developing 1-Bit Matrix Completion

Matrix completion has a long-time history of usage as the core technique of recommender systems. In particular, 1-bit matrix completion, which considers the prediction as a ``Recommended'' or ``Not Recommended'' question, has proved its significance and validity in the field. However, while customers and products aggregate into interacted clusters, state-of-the-art model-based 1-bit recommender systems do not take the consideration of grouping bias. To tackle the gap, this paper introduced Group-Specific 1-bit Matrix Completion (GS1MC) by first-time consolidating group-specific effects into 1-bit recommender systems under the low-rank latent variable framework. Additionally, to empower GS1MC even when grouping information is unobtainable, Cluster Developing Matrix Completion (CDMC) was proposed by integrating the sparse subspace clustering technique into GS1MC. Namely, CDMC allows clustering users/items and to leverage their group effects into matrix completion at the same time. Experiments on synthetic and real-world data show that GS1MC outperforms the current 1-bit matrix completion methods. Meanwhile, it is compelling that CDMC can successfully capture items' genre features only based on sparse binary user-item interactive data. Notably, GS1MC provides a new insight to incorporate and evaluate the efficacy of clustering methods while CDMC can be served as a new tool to explore unrevealed social behavior or market phenomenon.Comment: 16 Page

Tractable Clustering of Data on the Curve Manifold

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function. In this paper we propose a tractable method to cluster functional data or curves by adapting the Euclidean Low-Rank Representation (LRR) to the curve manifold. Experimental evaluation on synthetic and real data reveals that this method massively outperforms prior clustering methods in both speed and accuracy when clustering functional data.Comment: arXiv admin note: substantial text overlap with arXiv:1601.0073