Nearness to Local Subspace Algorithm for Subspace and Motion Segmentation

There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. This paper presents a clustering algorithm for high dimensional data that comes from a union of lower dimensional subspaces of equal and known dimensions. Such cases occur in many data clustering problems, such as motion segmentation and face recognition. The algorithm is reliable in the presence of noise, and applied to the Hopkins 155 Dataset, it generates the best results to date for motion segmentation. The two motion, three motion, and overall segmentation rates for the video sequences are 99.43%, 98.69%, and 99.24%, respectively

Nonlinear subspace clustering by functional link neural networks

Nonlinear subspace clustering based on a feed-forward neural network has been demonstrated to provide better clustering accuracy than some advanced subspace clustering algorithms. While this approach demonstrates impressive outcomes, it involves a balance between effectiveness and computational cost. In this study, we employ a functional link neural network to transform data samples into a nonlinear domain. Subsequently, we acquire a self-representation matrix through a learning mechanism that builds upon the mapped samples. As the functional link neural network is a single-layer neural network, our proposed method achieves high computational efficiency while ensuring desirable clustering performance. By incorporating the local similarity regularization to enhance the grouping effect, our proposed method further improves the quality of the clustering results. Additionally, we introduce a convex combination subspace clustering scheme, which combining a linear subspace clustering method with the functional link neural network subspace clustering approach. This combination approach allows for a dynamic balance between linear and nonlinear representations. Extensive experiments confirm the advancement of our methods. The source code will be released on https://lshi91.github.io/ soon

Community detection method based on mixed-norm sparse subspace clustering

Community or group is an important structure in disciplines such as social networks, biology gene expression, and physics systems. Community detections for different types of networks have attracted considerable interest. However, it is still challenging to find meaningful community structures in various networks. In particular, accurate community description and implementation of effective detection algorithms with huge datasets are still not solved. In this paper, we present a novel community detection algorithm based on the theory of sparse subspace clustering (SSC) with mixed-norm constraints. Inspired by the sparse representation of subspace, each community in a given network can span a subspace in some similarity measure space. If the basis of subspaces can be solved, all of the nodes can be represented as a linear combination of the nodes that span the same subspace. By introducing a novel mixed-norm constraint in SCC, the connections of nodes among different communities are modeled as noise to improve the clustering accuracy. The formulation of the basis of subspaces is derived from the self-representation property of data by using SSC. Then, the alternating directions method of multipliers (ADMM) framework is used to solve the formulation. Finally, communities are detected by subspace clustering method. The proposed method is compared with state-of-the-art algorithms on synthetic networks and real-world networks. The experimental results show the effectiveness of the proposed algorithm in accurately describing the community. The results also show that the mixed-norm SSC is a practical approach for detecting communities in huge datasets

Subspace Segmentation And High-Dimensional Data Analysis

This thesis developed theory and associated algorithms to solve subspace segmentation problem. Given a set of data W={w_1,...,w_N} in R^D that comes from a union of subspaces, we focused on determining a nonlinear model of the form U={S_i}_{i in I}, where S_i is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our first approach is based on the binary reduced row echelon form of data matrix. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace S_i. We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise. Our second approach is based on nearness to local subspaces approach and it can handle noise effectively, but it works only in special cases of the general subspace segmentation problem (i.e., subspaces of equal and known dimensions). Our approach is based on the computation of a binary similarity matrix for the data points. A local subspace is first estimated for each data point. Then, a distance matrix is generated by computing the distances between the local subspaces and points. The distance matrix is converted to the similarity matrix by applying a data-driven threshold. The problem is then transformed to segmentation of subspaces of dimension 1 instead of subspaces of dimension d. The algorithm was applied to the Hopkins 155 Dataset and generated the best results to date