Low-rank Representation Model Research Articles

Bayesian Low-Rank and Sparse Nonlinear Representation for Manifold Clustering

Linear representation usually used by the optimization model about low-rankness and sparsity limits their applications to some extent. In this paper, we propose Bayesian low-rank and sparse nonlinear representation (BLSN) model exploiting nonlinear representation. Different from the optimization model, BLSN can be solved by traditional algorithm in Bayesian statistics easily without knowing the explicit mapping by kernel trick. Moreover, it can learn the parameters adaptively to choose the low-rank and sparse properties and also provides a way to enforce more properties on one quantity in a Bayesian model. Based on the observation that the data points drawn from a union of manifolds may gain more meaningful linear structure after a nonlinear mapping, we apply BLSN for manifold clustering. It can handle different problems by constructing various kernels. With respect to the case of linear manifold, known as subspace segmentation, we propose a kernel by the Veronese mapping. In addition, we also design the kernel matrices for the case of nonlinear manifold. Experimental results confirm the effectiveness and the potential of our model for manifold clustering.

Dual Graph Regularized Latent Low-Rank Representation for Subspace Clustering.

Low-rank representation (LRR) has received considerable attention in subspace segmentation due to its effectiveness in exploring low-dimensional subspace structures embedded in data. To preserve the intrinsic geometrical structure of data, a graph regularizer has been introduced into LRR framework for learning the locality and similarity information within data. However, it is often the case that not only the high-dimensional data reside on a non-linear low-dimensional manifold in the ambient space, but also their features lie on a manifold in feature space. In this paper, we propose a dual graph regularized LRR model (DGLRR) by enforcing preservation of geometric information in both the ambient space and the feature space. The proposed method aims for simultaneously considering the geometric structures of the data manifold and the feature manifold. Furthermore, we extend the DGLRR model to include non-negative constraint, leading to a parts-based representation of data. Experiments are conducted on several image data sets to demonstrate that the proposed method outperforms the state-of-the-art approaches in image clustering.

Learning Balanced and Unbalanced Graphs via Low-Rank Coding

Graphs have been widely applied in modeling the relationships and structures in real-world applications. Graph construction is the most critical part in these models, while how to construct an effective graph is still an open problem. In this paper, we propose a novel approach to graph construction based on two observations. First, by virtue of recent advances in low-rank subspace recovery, the similarity between every two samples evaluated in the low-rank code space is more robust than that in the sample space. Second, a sparse and balanced graph can greatly increase the performance of learning tasks, such as label propagation in graph based semi-supervised learning. The <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula> -NN sparsification can provide fast solutions to constructing unbalanced sparse graphs, and <inline-formula><tex-math notation="LaTeX">$b$</tex-math></inline-formula> -matching constraint is a necessary route for generating balanced graphs. These observations motivate us to jointly learn the low-rank codes and balanced (or unbalanced) graph simultaneously. In particular, two non-convex models are built by incorporating <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula> -NN constraint and <inline-formula><tex-math notation="LaTeX">$b$</tex-math></inline-formula> -matching constraint into the low-rank representation model, respectively. We design a majorization-minimization augmented Lagrange multiplier (MM-ALM) algorithm to solve the proposed models. Extensive experimental results on four image databases demonstrate the superiority of our graphs over several state-of-the-art graphs in data clustering, transductive and inductive semi-supervised learning.

Nonlinear low‐rank representation on Stiefel manifolds

Recently, the low-rank representation (LRR) has been widely used in computer vision and pattern recognition with great success owing to its effectiveness and robustness for data clustering. However, the traditional LRR mainly focuses on the data from Euclidean space and is not directly applicable to manifold-valued data. A way to extend the LRR model from Euclidean space to the Stiefel manifold, by incorporating the intrinsic geometry of the manifold, is proposed. Under LRR, an appropriate affinity matrix for data on the Stiefel manifold can be learned; subsequently data clustering can be efficiently performed on the manifold. Experiments on several directional datasets demonstrate its superior performance on clustering compared with the state-of-the-art approaches.

Small infrared target detection based on low-rank and sparse representation

The method by which to obtain the correct detection result for infrared small targets is an important and challenging issue in infrared applications. In this paper, a low-rank and sparse representation (LRSR) model is proposed. This model can describe the specific structure of noise data effectively by utilizing sparse representation theory on the basis of low-rank matrix representation. In addition, LRSR based infrared small target detection algorithm is presented. First, a two-dimensional Gaussian model is used to produce the atoms that construct over-complete target dictionary. Then, the reset image data matrix is decomposed by the LRSR model to obtain the background, noise and target components of the image. Finally, the target position can be determined by threshold processing for the target component data. The experimental results in single objective frame, multi-objective image sequences, and strong noise background conditions demonstrate that the proposed method not only has high detection performance in effectively reducing the false alarm rate but also has strong robustness against noise interference.

Low-rank 3D mesh segmentation and labeling with structure guiding

Semantic mesh segmentation and labeling is a fundamental problem in graphics. Conventional data-driven approaches usually employ a tedious offline pre-training process. Moreover, the number and especially the quality of the manually labeled examples challenge such strategies. In this paper, we develop a low-rank representation model with structure guiding to address these problems. The pre-training step is successfully eliminated and a test mesh can be labeled just using a few examples. As consistently labeling a large amount of meshes manually is a tedious procedure accompanied by inevitable mislabelings, our method is indeed more suitable for semantic mesh segmentation and labeling in real situations. In additional, by introducing the guiding from geometric similarity and labeling structure, and the robust ℓ2,1 norm, our method generates correct labeling, even when the set of given examples contains multiple object categories or mislabeled meshes. Experimental results on the Princeton Segmentation Benchmark show that our approach outperforms the existing learning based methods.