Many state-of-the-art subspace clustering methods follow a two-step process by first constructing an affinity matrix between data points and then applying spectral clustering to this affinity. Most of the research into these methods focuses on the first step of generating the affinity, which often exploits the self-expressive property of linear subspaces, with little consideration typically given to the spectral clustering step that produces the final clustering. Moreover, existing methods often obtain the final affinity that is used in the spectral clustering step by applying ad-hoc or arbitrarily chosen postprocessing steps to the affinity generated by a self-expressive clustering formulation, which can have a significant impact on the overall clustering performance. In this work, we unify these two steps by learning both a self-expressive representation of the data and an affinity matrix that is well-normalized for spectral clustering. In our proposed models, we constrain the affinity matrix to be doubly stochastic, which results in a principled method for affinity matrix normalization while also exploiting known benefits of doubly stochastic normalization in spectral clustering. We develop a general framework and derive two models: one that jointly learns the self-expressive representation along with the doubly stochastic affinity, and one that sequentially solves for one then the other. Furthermore, we leverage sparsity in the problem to develop a fast active-set method for the sequential solver that enables efficient computation on large datasets. Experiments show that our method achieves state-of-the-art subspace clustering performance on many common datasets in computer vision.
University of Electronic Science and Technology of China（电子科技大学）
Finding a suitable data representation for a specific task has been shown to be crucial in many applications. The success of subspace clustering depends on the assumption that the data can be separated into different subspaces. However, this simple assumption does not always hold since the raw data might not be separable into subspaces. To recover the ``clustering-friendly'' representation and facilitate the subsequent clustering, we propose a graph filtering approach by which a smooth representation is achieved. Specifically, it injects graph similarity into data features by applying a low-pa…
Hyperspectral image (HSI) clustering is a challenging task due to the high complexity of HSI data. Subspace clustering has been proven to be powerful for exploiting the intrinsic relationship between data points. Despite the impressive performance in the HSI clustering, traditional subspace clustering methods often ignore the inherent structural information among data. In this paper, we revisit the subspace clustering with graph convolution and present a novel subspace clustering framework called Graph Convolutional Subspace Clustering (GCSC) for robust HSI clustering. Specifically, the framew…
Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the prominent subspace clustering algorithms rely on the representation of the data points as linear combinations of other data points, which is known as a self-expressive representation. To overcome the restrictive linearity assumption, numerous nonlinear approaches were proposed to extend successful subspace clustering approaches to data on a union of nonlinear m…
Mathematical Institute for Data Science（数据科学数学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of California, Berkeley（加州大学伯克利分校）
Subspace clustering is an unsupervised clustering technique designed to cluster data that is supported on a union of linear subspaces, with each subspace defining a cluster with dimension lower than the ambient space. Many existing formulations for this problem are based on exploiting the self-expressive property of linear subspaces, where any point within a subspace can be represented as linear combination of other points within the subspace. To extend this approach to data supported on a union of non-linear manifolds, numerous studies have proposed learning an embedding of the original data …
Algebraic Subspace Clustering (ASC) is a simple and elegant method based on polynomial fitting and differentiation for clustering noiseless data drawn from an arbitrary union of subspaces. In practice, however, ASC is limited to equi-dimensional subspaces because the estimation of the subspace dimension via algebraic methods is sensitive to noise. This paper proposes a new ASC algorithm that can handle noisy data drawn from subspaces of arbitrary dimensions. The key ideas are (1) to construct, at each point, a decreasing sequence of subspaces containing the subspace passing through that point;…
University of Electronic Science and Technology of China（电子科技大学）；Trusted Cloud Computing and Big Data Key Laboratory of Sichuan Province（四川省可信云计算与大数据重点实验室）
In recent years, multi-view subspace clustering has achieved impressive performance due to the exploitation of complementary imformation across multiple views. However, multi-view data can be very complicated and are not easy to cluster in real-world applications. Most existing methods operate on raw data and may not obtain the optimal solution. In this work, we propose a novel multi-view clustering method named smoothed multi-view subspace clustering (SMVSC) by employing a novel technique, i.e., graph filtering, to obtain a smooth representation for each view, in which similar data points hav…
Deep Subspace Clustering Networks (DSC) provide an efficient solution to the problem of unsupervised subspace clustering by using an undercomplete deep auto-encoder with a fully-connected layer to exploit the self expressiveness property. This method uses undercomplete representations of the input data which makes it not so robust and more dependent on pre-training. To overcome this, we propose a simple yet efficient alternative method - Overcomplete Deep Subspace Clustering Networks (ODSC) where we use overcomplete representations for subspace clustering. In our proposed method, we fuse the f…
Sparse subspace clustering methods, such as Sparse Subspace Clustering (SSC) \cite{ElhamifarV13} and $\ell^{1}$-graph \cite{YanW09,ChengYYFH10}, are effective in partitioning the data that lie in a union of subspaces. Most of those methods use $\ell^{1}$-norm or $\ell^{2}$-norm with thresholding to impose the sparsity of the constructed sparse similarity graph, and certain assumptions, e.g. independence or disjointness, on the subspaces are required to obtain the subspace-sparse representation, which is the key to their success. Such assumptions are not guaranteed to hold in practice and they …
The University of Sydney（悉尼大学）；Fudan University（复旦大学）
The neural ordinary differential equation (neural ODE) model has attracted increasing attention in time series analysis for its capability to process irregular time steps, i.e., data are not observed over equally-spaced time intervals. In multi-dimensional time series analysis, a task is to conduct evolutionary subspace clustering, aiming at clustering temporal data according to their evolving low-dimensional subspace structures. Many existing methods can only process time series with regular time steps while time series are unevenly sampled in many situations such as missing data. In this pap…
School of IoT Engineering, Jiangnan University（江南大学物联网工程学院）
Recently, sparse subspace clustering has been a valid tool to deal with high-dimensional data. There are two essential steps in the framework of sparse subspace clustering. One is solving the coefficient matrix of data, and the other is constructing the affinity matrix from the coefficient matrix, which is applied to the spectral clustering. This paper investigates the factors which affect clustering performance from both clustering accuracy and stability of the approaches based on existing algorithms. We select four methods to solve the coefficient matrix and use four different ways to constr…
Shamoon College of Engineering, Department of Electrical Engineering（谢幕工程学院，电气工程系）；Sapir Academic College, Department of Engineering（萨皮尔学术学院，工程系）；Afeka Tel-Aviv Academic College of Engineering, School of Electrical Engineering（阿法卡特拉维学术工程学院，电气工程学院）；Avignon University, LIA（阿维尼翁大学，LIA）
Many clustering problems in computer vision and other contexts are also classification problems, where each cluster shares a meaningful label. Subspace clustering algorithms in particular are often applied to problems that fit this description, for example with face images or handwritten digits. While it is straightforward to request human input on these datasets, our goal is to reduce this input as much as possible. We present a pairwise-constrained clustering algorithm that actively selects queries based on the union-of-subspaces model. The central step of the algorithm is in querying points…
The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
Subspace clustering (SC) aims to cluster data lying in a union of low-dimensional subspaces. Usually, SC learns an affinity matrix and then performs spectral clustering. Both steps suffer from high time and space complexity, which leads to difficulty in clustering large datasets. This paper presents a method called k-Factorization Subspace Clustering (k-FSC) for large-scale subspace clustering. K-FSC directly factorizes the data into k groups via pursuing structured sparsity in the matrix factorization model. Thus, k-FSC avoids learning affinity matrix and performing eigenvalue decomposition, …
We propose Deep Closed-Form Subspace Clustering (DCFSC), a new embarrassingly simple model for subspace clustering with learning non-linear mapping. Compared with the previous deep subspace clustering (DSC) techniques, our DCFSC does not have any parameters at all for the self-expressive layer. Instead, DCFSC utilizes the implicit data-driven self-expressive layer derived from closed-form shallow auto-encoder. Moreover, DCFSC also has no complicated optimization scheme, unlike the other subspace clustering methods. With its extreme simplicity, DCFSC has significant memory-related benefits over…
University of California, Berkeley（加州大学伯克利分校）；Beijing University of Posts and Telecommunications（北京邮电大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really…
South China Agricultural University（华南农业大学）；Sun Yat-sen University（中山大学）
Multi-view subspace clustering aims to discover the hidden subspace structures from multiple views for robust clustering, and has been attracting considerable attention in recent years. Despite significant progress, most of the previous multi-view subspace clustering algorithms are still faced with two limitations. First, they usually focus on the consistency (or commonness) of multiple views, yet often lack the ability to capture the cross-view inconsistencies in subspace representations. Second, many of them overlook the local structures of multiple views and cannot jointly leverage multiple…
Delhi Technical University（德里技术大学）；Indraprastha Institute of Information Technology Delhi（德里因陀罗普拉斯塔信息技术学院）
Subspace clustering techniques have shown promise in hyperspectral image segmentation. The fundamental assumption in subspace clustering is that the samples belonging to different clusters/segments lie in separable subspaces. What if this condition does not hold? We surmise that even if the condition does not hold in the original space, the data may be nonlinearly transformed to a space where it will be separable into subspaces. In this work, we propose a transformation based on the tenets of deep dictionary learning (DDL). In particular, we incorporate the sparse subspace clustering (SSC) los…
Subspace clustering is the classical problem of clustering a collection of data samples that approximately lie around several low-dimensional subspaces. The current state-of-the-art approaches for this problem are based on the self-expressive model which represents the samples as linear combination of other samples. However, these approaches require sufficiently well-spread samples for accurate representation which might not be necessarily accessible in many applications. In this paper, we shed light on this commonly neglected issue and argue that data distribution within each subspace plays a…
A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the literature to partition such high-dimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models…
China University of Petroleum, Beijing（中国石油大学（北京））
In this paper, we propose a novel Attentive Multi-View Deep Subspace Nets (AMVDSN), which deeply explores underlying consistent and view-specific information from multiple views and fuse them by considering each view's dynamic contribution obtained by attention mechanism. Unlike most multi-view subspace learning methods that they directly reconstruct data points on raw data or only consider consistency or complementarity when learning representation in deep or shallow space, our proposed method seeks to find a joint latent representation that explicitly considers both consensus and view-specif…
Matrix rank minimization problem is in general NP-hard. The nuclear norm is used to substitute the rank function in many recent studies. Nevertheless, the nuclear norm approximation adds all singular values together and the approximation error may depend heavily on the magnitudes of singular values. This might restrict its capability in dealing with many practical problems. In this paper, an arctangent function is used as a tighter approximation to the rank function. We use it on the challenging subspace clustering problem. For this nonconvex minimization problem, we develop an effective optim…
This paper proposes a novel deep subspace clustering approach which uses convolutional autoencoders to transform input images into new representations lying on a union of linear subspaces. The first contribution of our work is to insert multiple fully-connected linear layers between the encoder layers and their corresponding decoder layers to promote learning more favorable representations for subspace clustering. These connection layers facilitate the feature learning procedure by combining low-level and high-level information for generating multiple sets of self-expressive and informative re…
In this paper, we explore the problem of deep multi-view subspace clustering framework from an information-theoretic point of view. We extend the traditional information bottleneck principle to learn common information among different views in a self-supervised manner, and accordingly establish a new framework called Self-supervised Information Bottleneck based Multi-view Subspace Clustering (SIB-MSC). Inheriting the advantages from information bottleneck, SIB-MSC can learn a latent space for each view to capture common information among the latent representations of different views by removin…
Subspace clustering methods based on $\ell_1$, $\ell_2$ or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, $\ell_2…
Portland State University（波特兰州立大学）；University of Pennsylvania（宾夕法尼亚大学）；University of California, Berkeley（加州大学伯克利分校）；University of Michigan（密歇根大学）
Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees, or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the K-subspaces (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble K-subspaces (EKSS), forms a co-association matrix whose (i,j)th entry is th…
University of Cambridge（剑桥大学）；Lancaster University（兰卡斯特大学）
Spectral-based subspace clustering methods have proved successful in many challenging applications such as gene sequencing, image recognition, and motion segmentation. In this work, we first propose a novel spectral-based subspace clustering algorithm that seeks to represent each point as a sparse convex combination of a few nearby points. We then extend the algorithm to constrained clustering and active learning settings. Our motivation for developing such a framework stems from the fact that typically either a small amount of labelled data is available in advance; or it is possible to label …
Beijing University of Posts and Telecommunications（北京邮电大学）；University of California, Berkeley（加州大学伯克利分校）
State-of-the-art subspace clustering methods are based on self-expressive model, which represents each data point as a linear combination of other data points. By enforcing such representation to be sparse, sparse subspace clustering is guaranteed to produce a subspace-preserving data affinity where two points are connected only if they are from the same subspace. On the other hand, however, data points from the same subspace may not be well-connected, leading to the issue of over-segmentation. We introduce dropout to address the issue of over-segmentation, which is based on randomly dropping …
Anchor-based Multi-view Subspace Clustering with Hierarchical Feature Descent
College of Computer, National University of Defense Technology（国防科技大学计算机学院）；Intelligent Game and Decision Lab（智能游戏与决策实验室）；College of Intelligence Science and Technology, National University of Defense Technology（国防科技大学智能科学学院）
Indraprastha Institute of Information Technology（因德拉普拉斯信息技术学院）；CentraleSupelec（中央高等电力学院）
Subspace clustering assumes that the data is sepa-rable into separate subspaces. Such a simple as-sumption, does not always hold. We assume that, even if the raw data is not separable into subspac-es, one can learn a representation (transform coef-ficients) such that the learnt representation is sep-arable into subspaces. To achieve the intended goal, we embed subspace clustering techniques (locally linear manifold clustering, sparse sub-space clustering and low rank representation) into transform learning. The entire formulation is jointly learnt; giving rise to a new class of meth-ods called…
In this paper we present a new model and an algorithm for unsupervised clustering of 2-D data such as images. We assume that the data comes from a union of multilinear subspaces (UOMS) model, which is a specific structured case of the much studied union of subspaces (UOS) model. For segmentation under this model, we develop Multilinear Subspace Clustering (MSC) algorithm and evaluate its performance on the YaleB and Olivietti image data sets. We show that MSC is highly competitive with existing algorithms employing the UOS model in terms of clustering performance while enjoying improvement in …
State-of-the-art algorithms for sparse subspace clustering perform spectral clustering on a similarity matrix typically obtained by representing each data point as a sparse combination of other points using either basis pursuit (BP) or orthogonal matching pursuit (OMP). BP-based methods are often prohibitive in practice while the performance of OMP-based schemes are unsatisfactory, especially in settings where data points are highly similar. In this paper, we propose a novel algorithm that exploits an accelerated variant of orthogonal least-squares to efficiently find the underlying subspaces.…
South China Agricultural University（华南农业大学）；Pazhou Lab（琶洲实验室）；Sun Yat-sen University（中山大学）；Guangdong Key Laboratory of Information Security Technology（广东省信息安全技术重点实验室）；Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education（教育部机器智能与先进计算重点实验室）；Nanyang Technological University（南洋理工大学）
The rapid emergence of high-dimensional data in various areas has brought new challenges to current ensemble clustering research. To deal with the curse of dimensionality, recently considerable efforts in ensemble clustering have been made by means of different subspace-based techniques. However, besides the emphasis on subspaces, rather limited attention has been paid to the potential diversity in similarity/dissimilarity metrics. It remains a surprisingly open problem in ensemble clustering how to create and aggregate a large population of diversified metrics, and furthermore, how to jointly…
Most existing approaches address multi-view subspace clustering problem by constructing the affinity matrix on each view separately and afterwards propose how to extend spectral clustering algorithm to handle multi-view data. This paper presents an approach to multi-view subspace clustering that learns a joint subspace representation by constructing affinity matrix shared among all views. Relying on the importance of both low-rank and sparsity constraints in the construction of the affinity matrix, we introduce the objective that balances between the agreement across different views, while at …
Discovering and clustering subspaces in high-dimensional data is a fundamental problem of machine learning with a wide range of applications in data mining, computer vision, and pattern recognition. Earlier methods divided the problem into two separate stages of finding the similarity matrix and finding clusters. Similar to some recent works, we integrate these two steps using a joint optimization approach. We make the following contributions: (i) we estimate the reliability of the cluster assignment for each point before assigning a point to a subspace. We group the data points into two group…
In subspace clustering, a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces. This paper presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on their relative novelties. We present two frameworks in which the idea of innovation pursuit is used to distinguish the subspaces. Underlying the first framework is an iterative method that finds the subspaces consecutively by solving a series of simple linear optimization problems, …
China University of Petroleum, Beijing Campus (CUP)（中国石油大学（北京））
Multi-view learning can cover all features of data samples more comprehensively, so multi-view learning has attracted widespread attention. Traditional subspace clustering methods, such as sparse subspace clustering (SSC) and low-ranking subspace clustering (LRSC), cluster the affinity matrix for a single view, thus ignoring the problem of fusion between views. In our article, we propose a new Multiview Subspace Adaptive Learning based on Attention and Autoencoder (MSALAA). This method combines a deep autoencoder and a method for aligning the self-representations of various views in Multi-view…
Beijing University of Posts and Telecommunications（北京邮电大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Sparse subspace clustering (SSC) is a state-of-the-art method for segmenting a set of data points drawn from a union of subspaces into their respective subspaces. It is now well understood that SSC produces subspace-preserving data affinity under broad geometric conditions but suffers from a connectivity issue. In this paper, we develop a novel geometric analysis for a variant of SSC, named affine SSC (ASSC), for the problem of clustering data from a union of affine subspaces. Our contributions include a new concept called affine independence for capturing the arrangement of a collection of af…
Subspace clustering aims to cluster unlabeled data that lies in a union of low-dimensional linear subspaces. Deep subspace clustering approaches based on auto-encoders have become very popular to solve subspace clustering problems. However, the training of current deep methods converges slowly, which is much less efficient than traditional approaches. We propose a Residual Encoder-Decoder network for deep Subspace Clustering (RED-SC), which symmetrically links convolutional and deconvolutional layers with skip-layer connections, with which the training converges much faster. We use a self-expr…
Subspace clustering refers to the problem of clustering high-dimensional data into a union of low-dimensional subspaces. Current subspace clustering approaches are usually based on a two-stage framework. In the first stage, an affinity matrix is generated from data. In the second one, spectral clustering is applied on the affinity matrix. However, the affinity matrix produced by two-stage methods cannot fully reveal the similarity between data points from the same subspace (intra-subspace similarity), resulting in inaccurate clustering. Besides, most approaches fail to solve large-scale cluste…
This letter presents a new spectral-clustering-based approach to the subspace clustering problem. Underpinning the proposed method is a convex program for optimal direction search, which for each data point d finds an optimal direction in the span of the data that has minimum projection on the other data points and non-vanishing projection on d. The obtained directions are subsequently leveraged to identify a neighborhood set for each data point. An alternating direction method of multipliers framework is provided to efficiently solve for the optimal directions. The proposed method is shown to…
Concepts are key building blocks of higher level human understanding. Explainable AI (XAI) methods have shown tremendous progress in recent years, however, local attribution methods do not allow to identify coherent model behavior across samples and therefore miss this essential component. In this work, we study concept-based explanations and put forward a new definition of concepts as low-dimensional subspaces of hidden feature layers. We novelly apply sparse subspace clustering to discover these concept subspaces. Moving forward, we derive insights from concept subspaces in terms of localize…
Nanjing University of Science and Technology（南京理工大学）；Brandeis University（布兰迪斯大学）；Adobe Research（Adobe研究院）；Northeastern University（东北大学）
This paper studies the large-scale subspace clustering (LSSC) problem with million data points. Many popular subspace clustering methods cannot directly handle the LSSC problem although they have been considered as state-of-the-art methods for small-scale data points. A basic reason is that these methods often choose all data points as a big dictionary to build huge coding models, which results in a high time and space complexity. In this paper, we develop a learnable subspace clustering paradigm to efficiently solve the LSSC problem. The key idea is to learn a parametric function to partition…
College of Computer Science & Technology, Nanjing University of Aeronautics & Astronautics（南京航空航天大学计算机科学与技术学院）；Collaborative Innovation Center of Novel Software Technology and Industrialization（新型软件技术与产业化协同创新中心）
Nowadays, multi-view clustering has attracted more and more attention. To date, almost all the previous studies assume that views are complete. However, in reality, it is often the case that each view may contain some missing instances. Such incompleteness makes it impossible to directly use traditional multi-view clustering methods. In this paper, we propose a Doubly Aligned Incomplete Multi-view Clustering algorithm (DAIMC) based on weighted semi-nonnegative matrix factorization (semi-NMF). Specifically, on the one hand, DAIMC utilizes the given instance alignment information to learn a comm…
Subspace clustering is the problem of clustering data points into a union of low-dimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is …
School of Software Engineering, Xi’an Jiaotong University（西安交通大学软件学院）
This study investigates the problem of multi-view subspace clustering, the goal of which is to explore the underlying grouping structure of data collected from different fields or measurements. Since data do not always comply with the linear subspace models in many real-world applications, most existing multi-view subspace clustering methods that based on the shallow linear subspace models may fail in practice. Furthermore, underlying graph information of multi-view data is always ignored in most existing multi-view subspace clustering methods. To address aforementioned limitations, we propose…
Ruder Bošković Institute（鲁杰尔·博什科维奇研究所）；ETH Zürich, Swiss Federal Institute of Technology（苏黎世联邦理工学院）
A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal multiplicative update rules to solve the subspace clustering problem. In nonlinear orthogonal NMF framework, we propose two subspace clustering algorithms, named kernel-based non-negative subspace clustering KNSC-Ncut and KNSC-Rcut and establish their connection with spectral norm…
Jiangnan University（江南大学）；University of California, Davis（加利福尼亚大学戴维斯分校）；The Hong Kong Polytechnic University（香港理工大学）
Subspace clustering (SC) is a promising clustering technology to identify clusters based on their associations with subspaces in high dimensional spaces. SC can be classified into hard subspace clustering (HSC) and soft subspace clustering (SSC). While HSC algorithms have been extensively studied and well accepted by the scientific community, SSC algorithms are relatively new but gaining more attention in recent years due to better adaptability. In the paper, a comprehensive survey on existing SSC algorithms and the recent development are presented. The SSC algorithms are classified systematic…
Beihang University（北京航空航天大学）；Griffith University（格里菲斯大学）；University of York, UK（英国约克大学）
Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficie…
School of Computer Science, Fudan University（复旦大学计算机科学学院）；University of Technology, Sydney（悉尼科技大学）
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 …
Southwest University（西南大学）；George Mason University（乔治梅森大学）；China University of Geosciences（中国地质大学）；Beijing University Of Civil Engineering and Architecture（北京建筑大学）
Multiple clustering aims at discovering diverse ways of organizing data into clusters. Despite the progress made, it's still a challenge for users to analyze and understand the distinctive structure of each output clustering. To ease this process, we consider diverse clusterings embedded in different subspaces, and analyze the embedding subspaces to shed light into the structure of each clustering. To this end, we provide a two-stage approach called MISC (Multiple Independent Subspace Clusterings). In the first stage, MISC uses independent subspace analysis to seek multiple and statistical ind…
Johns Hopkins University（约翰斯·霍普金斯大学）；Beijing University of Posts and Telecommunications（北京邮电大学）
State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；DePaul University（德保罗大学）
Clustering can be defined as the process of assembling objects into a number of groups whose elements are similar to each other in some manner. As a technique that is used in many domains, such as face clustering, plant categorization, image segmentation, document classification, clustering is considered one of the most important unsupervised learning problems. Scientists have surveyed this problem for years and developed different techniques that can solve it, such as k-means clustering. We analyze one of these techniques: a powerful clustering algorithm called Sparse Subspace Clustering. We …
