Scaled Simplex Representation for Subspace Clustering

The self-expressive property of data points, i.e., each data point can be linearly represented by the other data points in the same subspace, has proven effective in leading subspace clustering methods. Most self-expressive methods usually construct a feasible affinity matrix from a coefficient matrix, obtained by solving an optimization problem. However, the negative entries in the coefficient matrix are forced to be positive when constructing the affinity matrix via exponentiation, absolute symmetrization, or squaring operations. This consequently damages the inherent correlations among the data. Besides, the affine constraint used in these methods is not flexible enough for practical applications. To overcome these problems, in this paper, we introduce a scaled simplex representation (SSR) for subspace clustering problem. Specifically, the non-negative constraint is used to make the coefficient matrix physically meaningful, and the coefficient vector is constrained to be summed up to a scalar s<1 to make it more discriminative. The proposed SSR based subspace clustering (SSRSC) model is reformulated as a linear equality-constrained problem, which is solved efficiently under the alternating direction method of multipliers framework. Experiments on benchmark datasets demonstrate that the proposed SSRSC algorithm is very efficient and outperforms state-of-the-art subspace clustering methods on accuracy. The code can be found at https://github.com/csjunxu/SSRSC.Comment: Accepted by IEEE Transactions on Cybernetics. 13 pages, 9 figures, 10 tables. Code can be found at https://github.com/csjunxu/SSRS

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

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

Deep Sparse Subspace Clustering

In this paper, we present a deep extension of Sparse Subspace Clustering, termed Deep Sparse Subspace Clustering (DSSC). Regularized by the unit sphere distribution assumption for the learned deep features, DSSC can infer a new data affinity matrix by simultaneously satisfying the sparsity principle of SSC and the nonlinearity given by neural networks. One of the appealing advantages brought by DSSC is: when original real-world data do not meet the class-specific linear subspace distribution assumption, DSSC can employ neural networks to make the assumption valid with its hierarchical nonlinear transformations. To the best of our knowledge, this is among the first deep learning based subspace clustering methods. Extensive experiments are conducted on four real-world datasets to show the proposed DSSC is significantly superior to 12 existing methods for subspace clustering.Comment: The initial version is completed at the beginning of 201

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

Self-Supervised Convolutional Subspace Clustering Network

Subspace clustering methods based on data self-expression have become very popular for learning from data that lie in a union of low-dimensional linear subspaces. However, the applicability of subspace clustering has been limited because practical visual data in raw form do not necessarily lie in such linear subspaces. On the other hand, while Convolutional Neural Network (ConvNet) has been demonstrated to be a powerful tool for extracting discriminative features from visual data, training such a ConvNet usually requires a large amount of labeled data, which are unavailable in subspace clustering applications. To achieve simultaneous feature learning and subspace clustering, we propose an end-to-end trainable framework, called Self-Supervised Convolutional Subspace Clustering Network (S$^2$ConvSCN), that combines a ConvNet module (for feature learning), a self-expression module (for subspace clustering) and a spectral clustering module (for self-supervision) into a joint optimization framework. Particularly, we introduce a dual self-supervision that exploits the output of spectral clustering to supervise the training of the feature learning module (via a classification loss) and the self-expression module (via a spectral clustering loss). Our experiments on four benchmark datasets show the effectiveness of the dual self-supervision and demonstrate superior performance of our proposed approach.Comment: 10 pages, 2 figures, and 5 tables. This paper has been accepted by CVPR201

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

Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds

As a significant subspace clustering method, low rank representation (LRR) has attracted great attention in recent years. To further improve the performance of LRR and extend its applications, there are several issues to be resolved. The nuclear norm in LRR does not sufficiently use the prior knowledge of the rank which is known in many practical problems. The LRR is designed for vectorial data from linear spaces, thus not suitable for high dimensional data with intrinsic non-linear manifold structure. This paper proposes an extended LRR model for manifold-valued Grassmann data which incorporates prior knowledge by minimizing partial sum of singular values instead of the nuclear norm, namely Partial Sum minimization of Singular Values Representation (GPSSVR). The new model not only enforces the global structure of data in low rank, but also retains important information by minimizing only smaller singular values. To further maintain the local structures among Grassmann points, we also integrate the Laplacian penalty with GPSSVR. An effective algorithm is proposed to solve the optimization problem based on the GPSSVR model. The proposed model and algorithms are assessed on some widely used human action video datasets and a real scenery dataset. The experimental results show that the proposed methods obviously outperform other state-of-the-art methods.Comment: Submitting to ACM Transactions on Knowledge Discovery from Data with minor revisio

Deep Multimodal Subspace Clustering Networks

We present convolutional neural network (CNN) based approaches for unsupervised multimodal subspace clustering. The proposed framework consists of three main stages - multimodal encoder, self-expressive layer, and multimodal decoder. The encoder takes multimodal data as input and fuses them to a latent space representation. The self-expressive layer is responsible for enforcing the self-expressiveness property and acquiring an affinity matrix corresponding to the data points. The decoder reconstructs the original input data. The network uses the distance between the decoder's reconstruction and the original input in its training. We investigate early, late and intermediate fusion techniques and propose three different encoders corresponding to them for spatial fusion. The self-expressive layers and multimodal decoders are essentially the same for different spatial fusion-based approaches. In addition to various spatial fusion-based methods, an affinity fusion-based network is also proposed in which the self-expressive layer corresponding to different modalities is enforced to be the same. Extensive experiments on three datasets show that the proposed methods significantly outperform the state-of-the-art multimodal subspace clustering methods

Groupwise Constrained Reconstruction for Subspace Clustering

Reconstruction based subspace clustering methods compute a self reconstruction matrix over the samples and use it for spectral clustering to obtain the final clustering result. Their success largely relies on the assumption that the underlying subspaces are independent, which, however, does not always hold in the applications with increasing number of subspaces. In this paper, we propose a novel reconstruction based subspace clustering model without making the subspace independence assumption. In our model, certain properties of the reconstruction matrix are explicitly characterized using the latent cluster indicators, and the affinity matrix used for spectral clustering can be directly built from the posterior of the latent cluster indicators instead of the reconstruction matrix. Experimental results on both synthetic and real-world datasets show that the proposed model can outperform the state-of-the-art methods.Comment: ICML201