Subspace Segmentation Problem Research Articles

Grouping Subspace Segmentation

This work studies the subspace segmentation problem. Given a set of data points which are drawn from a union of multiple subspaces. The goal of subspace segmentation is to cluster the data into the underlying subspaces from which the data are drawn from. The most recent works use spectral clustering based on the affinity matrix obtained by Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Least Squares Regression (LSR). They write each data point as a linear combination of all the data points, and use the representation coefficients to form the affinity matrix. LSR shows that it is more effective due to the grouping effect for data with high correlation. In this work, we propose the Grouping Subspace Segmentation (GSS) method by enhancing the grouping effect of correlated data points. The affinity graph is constructed to encode the local structure of data. Experiments on the real data sets show that the proposed method GSS is more effective than the state-of-the-art methods.

Recursive Sample Scaling Low-Rank Representation

The low-rank representation (LRR) method has recently gained enormous popularity due to its robust approach in solving the subspace segmentation problem, particularly those concerning corrupted data. In this paper, the recursive sample scaling low-rank representation (RSS-LRR) method is proposed. The advantage of RSS-LRR over traditional LRR is that a cosine scaling factor is further introduced, which imposes a penalty on each sample to minimize noise and outlier influence better. Specifically, the cosine scaling factor is a similarity measure learned to extract each sample’s relationship with the low-rank representation’s principal components in the feature space. In order words, the smaller the angle between an individual data sample and the low-rank representation’s principal components, the more likely it is that the data sample is clean. Thus, the proposed method can then effectively obtain a good low-rank representation influenced mainly by clean data. Several experiments are performed with varying levels of corruption on ORL, CMU PIE, COIL20, COIL100, and LFW in order to evaluate RSS-LRR’s effectiveness over state-of-the-art low-rank methods. The experimental results show that RSS-LRR consistently performs better than the compared methods in image clustering and classification tasks.

A Group Norm Regularized Factorization Model for Subspace Segmentation

Subspace segmentation assumes that data comes from the union of different subspaces and the purpose of segmentation is to partition the data into the corresponding subspace. Low-rank representation (LRR) is a classic spectral-type method for solving subspace segmentation problems, that is, one first obtains an affinity matrix by solving a LRR model and then performs spectral clustering for segmentation. This paper proposes a group norm regularized factorization model (GNRFM) inspired by the LRR model for subspace segmentation and then designs an Accelerated Augmented Lagrangian Method (AALM) algorithm to solve this model. Specifically, we adopt group norm regularization to make the columns of the factor matrix sparse, thereby achieving a purpose of low rank, which means no Singular Value Decompositions (SVD) are required and the computational complexity of each step is greatly reduced. We obtain affinity matrices by using different LRR models and then performing cluster testing on different sets of synthetic noisy data and real data, respectively. Compared with traditional models and algorithms, the proposed method is faster and more robust to noise, so the final clustering results are better. Moreover, the numerical results show that our algorithm converges fast and only requires approximately ten iterations.

Robust Subspace Segmentation by Self-Representation Constrained Low-Rank Representation

Low-rank representation (LRR) and its variants have been proved to be powerful tools for handling subspace segmentation problems. In this paper, we propose a new LRR-related algorithm, termed self-representation constrained low-rank presentation (SRLRR). SRLRR contains a self-representation constraint which is used to compel the obtained coefficient matrices can be reconstructed by themselves. An optimization algorithm for solving SRLRR problem is also proposed. Moreover, we present an alternative formulation of SRLRR so that SRLRR can be regarded as a kind of Laplacian regularized LRR. Consequently, the relationships and comparisons between SRLRR and some existing Laplacian regularized LRR-related algorithms have been discussed. Finally, subspace segmentation experiments conducted on both synthetic and real databases show that SRLRR dominates the related algorithms.

Similarity matrix framework for data from union of subspaces

This paper presents a framework for finding similarity matrices for the segmentation of data W = [ w 1 ⋯ w N ] ⊂ R D drawn from a union U = ⋃ i = 1 M S i of independent subspaces { S i } i = 1 M of dimensions { d i } i = 1 M . It is shown that any factorization of W = B P , where columns of B form a basis for data W and they also come from U , can be used to produce a similarity matrix Ξ W . In other words, Ξ W ( i , j ) ≠ 0 , when the columns w i and w j of W come from the same subspace, and Ξ W ( i , j ) = 0 , when the columns w i and w j of W come from different subspaces. Furthermore, Ξ W = Q d m a x , where d m a x = max ⁡ { d i } i = 1 M and Q ∈ R N × N with Q ( i , j ) = | P T P ( i , j ) | . It is shown that a similarity matrix obtained from the reduced row echelon form of W is a special case of the theory. It is also proven that the Shape Interaction Matrix defined as V V T , where W = U Σ V T is the skinny singular value decomposition of W , is not necessarily a similarity matrix. But, taking powers of its absolute value always generates a similarity matrix. An interesting finding of this research is that a similarity matrix can be obtained using a skeleton decomposition of W . First, a square sub-matrix A ∈ R r × r of W with the same rank r as W is found. Then, the matrix R corresponding to the rows of W that contain A is constructed. Finally, a power of the matrix P T P where P = A − 1 R provides a similarity matrix Ξ W . Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to other constructions of similarity matrices.

Study on Different Representation Methods for Subspace Segmentation

With many engineering and science application problems, we must deal with a lot of high-dimensional data, such as videos, images, web documents, text, etc. In the areas of computer vision, image processing and machine learning, high-dimensional data are widespread. However, it is very hard for obtaining meaningful learning and inference from these high-dimensional data directly, the computational complexity of highdimensional data is often exponential. However, under many conditions, highdimensional data lie in low-dimensional data corresponding to some classes of the data. Thus, finding the low-dimensional structure from the high-dimensional data is very important. The aim of subspace segmentation is to cluster data that lie in a union of lowdimensional subspaces. In recent years, based on the research of representation methods, many subspace segmentation algorithms appeared. Although these methods are all effective for handling subspace segmentation problems, they all have advantages and disadvantages. This paper focuses on the performance comparison of different subspace segmentation algorithms currently used in handling subspace segmentation problems and views other conventional methods that can be applied in this field.

A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation

The subspace segmentation problem is fundamental in many applications. The goal is to cluster data drawn from an unknown union of subspaces. In this paper we state the problem and describe its connection to other areas of mathematics and engineering. We then review the mathematical and algorithmic methods created to solve this problem and some of its particular cases. We also describe the problem of motion tracking in videos and its connection to the subspace segmentation problem and compare the various techniques for solving it.

Nonlinear Least Squares in ℝ N

Recent research and new paradigms in mathematics, engineering, and science assume nonlinear signal models of the form ℳ=∪i∈IVi consisting of a union of subspaces Vi instead of a single subspace ℳ=V. These models have been used in sampling and reconstruction of signals with finite rate of innovation, the Generalized Principle Component Analysis and the subspace segmentation problem in computer vision, and problems related to sparsity, compressed sensing, and dictionary design.

Optimal Non-Linear Models for Sparsity and Sampling

Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of ℝ N and to infinite dimensional shift-invariant spaces in L 2(ℝ d ). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.