In this paper, we analyse classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically with the number of nodes, guarantees in the static case can be extended to the dynamic case and yield improved error bounds when the DSBM is sufficiently smooth in time, that is, the communities do not change too much between two time steps. We improve over these results by drawing a new link between the sparsity and the smoothness of the DSBM: the more regular the DSBM is, the more sparse it can be, while still guaranteeing consistent recovery. In particular, a mild condition on the smoothness allows to treat the sparse case with bounded degree. We also extend these guarantees to the normalized Laplacian, and as a by-product of our analysis, we obtain to our knowledge the best spectral concentration bound available for the normalized Laplacian of matrices with independent Bernoulli entries.
There has been a surge in the number of large and flat data sets - data sets containing a large number of features and a relatively small number of observations - due to the growing ability to collect and store information in medical research and other fields. Hierarchical clustering is a widely used clustering tool. In hierarchical clustering, large and flat data sets may allow for a better coverage of clustering features (features that help explain the true underlying clusters) but, such data sets usually include a large fraction of noise features (non-clustering features) that may hide the …
In this paper we propose a mixture model, SparseMix, for clustering of sparse high dimensional binary data, which connects model-based with centroid-based clustering. Every group is described by a representative and a probability distribution modeling dispersion from this representative. In contrast to classical mixture models based on EM algorithm, SparseMix: -is especially designed for the processing of sparse data, -can be efficiently realized by an on-line Hartigan optimization algorithm, -is able to automatically reduce unnecessary clusters. We perform extensive experimental studies on va…
We consider the problem of clustering data points in high dimensions, i.e. when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimensions. The method we propose is a combination of a recent approach for learning parameters of a Gaussian mixture model and sparse linear discriminant analysis (LDA). In addition to cluster assignments, the method returns an estimate of the set of features relevant for clustering. …
Orthogonal Matching Pursuit (OMP) plays an important role in data science and its applications such as sparse subspace clustering and image processing. However, the existing OMP-based approaches lack of data adaptiveness so that the data cannot be represented well enough and may lose the accuracy. This paper proposes a novel approach to enhance the data-adaptive capability for OMP-based sparse subspace clustering. In our method a parameter selection process is developed to adjust the parameters based on the data distribution for information representation. Our theoretical analysis indicates th…
Courant Institute of Mathematical Sciences, NYU（纽约大学柯朗数学科学研究所）
We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size. Our theoretical analysis focuses on the model $X_i = z_i θ+ \varepsilon_i, ~z_i \in \{-1,1\}, ~\varepsilon_i \thicksim \mathcal{N}(0,I)$, which has two clusters with centres $θ$ and $-θ$. We provide a finite sample analysis of a new sparse clustering algorithm based on sparse PCA and show that it achieves the minimax optimal misclustering rate in the regime $\|θ\| \rightarrow \infty$. Our results require the sparsity to grow slow…
Meila (2018) introduces an optimization based method called the Sublevel Set method, to guarantee that a clustering is nearly optimal and "approximately correct" without relying on any assumptions about the distribution that generated the data. This paper extends the Sublevel Set method to the cost-based hierarchical clustering paradigm proposed by Dasgupta (2016).
NTT Advanced Technology Corporation（NTT先进技术公司）；The University of Electro-Communications（电气通信大学）
Sparse convex clustering is to cluster observations and conduct variable selection simultaneously in the framework of convex clustering. Although a weighted $L_1$ norm is usually employed for the regularization term in sparse convex clustering, its use increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering method based on the ideas of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures …
Information Security Lab, Institute of Big Data Technologies,\ Graduate School, Peking University
Sparse Subspace Clustering (SSC) is one of the most popular methods for clustering data points into their underlying subspaces. However, SSC may suffer from heavy computational burden. Orthogonal Matching Pursuit applied on SSC accelerates the computation but the trade-off is the loss of clustering accuracy. In this paper, we propose a noise-robust algorithm, Restricted Connection Orthogonal Matching Pursuit for Sparse Subspace Clustering (RCOMP-SSC), to improve the clustering accuracy and maintain the low computational time by restricting the number of connections of each data point during 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 …
New York University School of Medicine（纽约大学医学院）；Merck Research Laboratories（默克研究实验室）；University of Miami School of Business Administration（迈阿密大学工商管理学院）；New Jersey Institute of Technology（新泽西理工学院）
Convex clustering, a convex relaxation of k-means clustering and hierarchical clustering, has drawn recent attentions since it nicely addresses the instability issue of traditional nonconvex clustering methods. Although its computational and statistical properties have been recently studied, the performance of convex clustering has not yet been investigated in the high-dimensional clustering scenario, where the data contains a large number of features and many of them carry no information about the clustering structure. In this paper, we demonstrate that the performance of convex clustering co…
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 …
University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technical University of Munich（慕尼黑工业大学）
Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions …
The main contribution of the paper is a new approach to subspace clustering that is significantly more computationally efficient and scalable than existing state-of-the-art methods. The central idea is to modify the regression technique in sparse subspace clustering (SSC) by replacing the $\ell_1$ minimization with a generalization called Ordered Weighted $\ell_1$ (OWL) minimization which performs simultaneous regression and clustering of correlated variables. Using random geometric graph theory, we prove that OWL regression selects more points within each subspace, resulting in better cluster…
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…
Consider the problem of sparse clustering, where it is assumed that only a subset of the features are useful for clustering purposes. In the framework of the COSA method of Friedman and Meulman, subsequently improved in the form of the Sparse K-means method of Witten and Tibshirani, a natural and simpler hill-climbing approach is introduced. The new method is shown to be competitive with these two methods and others.
Department of Statistical Science, Duke University（杜克大学统计科学系）；Departments of Computational Medicine, Statistics, and Human Genetics, UCLA（加州大学洛杉矶分校计算医学、统计与人类遗传学系）
Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, sim…
We propose the Lasso Weighted $k$-means ($LW$-$k$-means) algorithm as a simple yet efficient sparse clustering procedure for high-dimensional data where the number of features ($p$) can be much larger compared to the number of observations ($n$). In the $LW$-$k$-means algorithm, we introduce a lasso-based penalty term, directly on the feature weights to incorporate feature selection in the framework of sparse clustering. $LW$-$k$-means does not make any distributional assumption of the given dataset and thus, induces a non-parametric method for feature selection. We also analytically investiga…
Subspace clustering methods based on expressing each data point as a linear combination of all other points in a dataset are popular unsupervised learning techniques. However, existing methods incur high computational complexity on large-scale datasets as they require solving an expensive optimization problem and performing spectral clustering on large affinity matrices. This paper presents an efficient approach to subspace clustering by selecting a small subset of the input data called landmarks. The resulting subspace clustering method in the reduced domain runs in linear time with respect t…
Cornell University（康奈尔大学）；University of Waterloo（滑铁卢大学）
We propose a clustering method that involves chaining four known techniques into a pipeline yielding an algorithm with stronger recovery guarantees than any of the four components separately. Given $n$ points in $\mathbb R^d$, the first component of our pipeline, which we call leapfrog distances, is reminiscent of density-based clustering, yielding an $n\times n$ distance matrix. The leapfrog distances are then translated to new embeddings using multidimensional scaling and spectral methods, two other known techniques, yielding new embeddings of the $n$ points in $\mathbb R^{d'}$, where $d'$ s…
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.…
MD Anderson Cancer Center（MD安德森癌症中心）；University of Pittsburgh（匹兹堡大学）；University of Maryland（马里兰大学）
Clustering with variable selection is a challenging yet critical task for modern small-n-large-p data. Existing methods based on sparse Gaussian mixture models or sparse K-means provide solutions to continuous data. With the prevalence of RNA-seq technology and lack of count data modeling for clustering, the current practice is to normalize count expression data into continuous measures and apply existing models with Gaussian assumption. In this paper, we develop a negative binomial mixture model with lasso or fused lasso gene regularization to cluster samples (small n) with high-dimensional g…
Microsoft Research（微软研究院）；University of California, Irvine（加州大学欧文分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We consider the problem of learning overcomplete dictionaries in the context of sparse coding, where each sample selects a sparse subset of dictionary elements. Our main result is a strategy to approximately recover the unknown dictionary using an efficient algorithm. Our algorithm is a clustering-style procedure, where each cluster is used to estimate a dictionary element. The resulting solution can often be further cleaned up to obtain a high accuracy estimate, and we provide one simple scenario where $\ell_1$-regularized regression can be used for such a second stage.
STOR-i Centre for Doctoral Training Lancaster University（兰卡斯特大学STOR-i博士培训中心）；Lancaster University（兰卡斯特大学）；Department of Mathematics and Statistics Lancaster University（兰卡斯特大学数学与统计系）；Department of Management Science Lancaster University（兰卡斯特大学管理科学系）；Data Science Campus Office for National Statistics（英国国家统计局数据科学园区办公室）；Lancaster, UK（英国兰卡斯特）；Newport, UK（英国纽波特）
In this paper we consider the problem of clustering collections of very short texts using subspace clustering. This problem arises in many applications such as product categorisation, fraud detection, and sentiment analysis. The main challenge lies in the fact that the vectorial representation of short texts is both high-dimensional, due to the large number of unique terms in the corpus, and extremely sparse, as each text contains a very small number of words with no repetition. We propose a new, simple subspace clustering algorithm that relies on linear algebra to cluster such datasets. Exper…
Graduate School of Informatics, Kyoto University（京都大学信息学院研究生院）
This paper introduces an elasticity reconstruction method based on local displacement observations of elastic bodies. Sparse reconstruction theory is applied to formulate the underdetermined inverse problems of elasticity reconstruction including unobserved areas. An online local clustering scheme called a superelement is proposed to reduce the number of dimensions of the optimization parameters. Alternating the optimization of element boundaries and elasticity parameters enables the elasticity distribution to be estimated with a higher spatial resolution. The simulation experiments show that …
University of Florida Informatics Institute（佛罗里达大学信息学研究所）；Rice University（莱斯大学）
Clustering is a ubiquitous problem in data science and signal processing. In many applications where we observe noisy signals, it is common practice to first denoise the data, perhaps using wavelet denoising, and then to apply a clustering algorithm. In this paper, we develop a sparse convex wavelet clustering approach that simultaneously denoises and discovers groups. Our approach utilizes convex fusion penalties to achieve agglomeration and group-sparse penalties to denoise through sparsity in the wavelet domain. In contrast to common practice which denoises then clusters, our method is a un…
Harbin University of Science and Technology（哈尔滨理工大学）
Density peaks clustering has become a nova of clustering algorithm because of its simplicity and practicality. However, there is one main drawback: it is time-consuming due to its high computational complexity. Herein, a density peaks clustering algorithm with sparse search and K-d tree is developed to solve this problem. Firstly, a sparse distance matrix is calculated by using K-d tree to replace the original full rank distance matrix, so as to accelerate the calculation of local density. Secondly, a sparse search strategy is proposed to accelerate the computation of relative-separation with …
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 …
We discuss a clustering method for Gaussian mixture model based on the sparse principal component analysis (SPCA) method and compare it with the IF-PCA method. We also discuss the dependent case where the covariance matrix $Σ$ is not necessarily diagonal.
Subspace sparse coding (SSC) algorithms have proven to be beneficial to clustering problems. They provide an alternative data representation in which the underlying structure of the clusters can be better captured. However, most of the research in this area is mainly focused on enhancing the sparse coding part of the problem. In contrast, we introduce a novel objective term in our proposed SSC framework which focuses on the separability of data points in the coding space. We also provide mathematical insights into how this local-separability term improves the clustering result of the SSC frame…
Neural networks are becoming increasingly popular in applications, but our mathematical understanding of their potential and limitations is still limited. In this paper, we further this understanding by developing statistical guarantees for sparse deep learning. In contrast to previous work, we consider different types of sparsity, such as few active connections, few active nodes, and other norm-based types of sparsity. Moreover, our theories cover important aspects that previous theories have neglected, such as multiple outputs, regularization, and l2-loss. The guarantees have a mild dependen…
Finnish Institute of Occupational Health（芬兰职业健康研究所）；Stockholm University（斯德哥尔摩大学）
Clustering is a widely used unsupervised learning method for finding structure in the data. However, the resulting clusters are typically presented without any guarantees on their robustness; slightly changing the used data sample or re-running a clustering algorithm involving some stochastic component may lead to completely different clusters. There is, hence, a need for techniques that can quantify the instability of the generated clusters. In this study, we propose a technique for quantifying the instability of a clustering solution and for finding robust clusters, termed core clusters, whi…
Université de Rouen（鲁昂大学）；INSA de Rouen（鲁昂国立应用科学学院）
To answer the existence of optimal swimmer learning/teaching strategies, this work introduces a two-level clustering in order to analyze temporal dynamics of motor learning in breaststroke swimming. Each level have been performed through Sparse Fisher-EM, a unsupervised framework which can be applied efficiently on large and correlated datasets. The induced sparsity selects key points of the coordination phase without any prior knowledge.
University of California Los Angeles（加利福尼亚大学洛杉矶分校）；University of California Davis（加利福尼亚大学戴维斯分校）；University of California at Davis（加利福尼亚大学戴维斯分校）
The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the NP-hard minimum ratio cut problem. In this paper we study the central question: when is spectral clustering able to find the global solution to the minimum ratio cut problem? First we provide a condition that naturally depends on the intra- and inter-cluster connectivities of a given partition under which we may certify that this partition is the solution to the minimum ratio cut problem. Then we dev…
GIPSA-lab（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔国立理工学院联合实验室）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Laboratoire d’Informatique de Grenoble (LIG)（格勒诺布尔信息学实验室）
This article considers spectral community detection in the regime of sparse networks with heterogeneous degree distributions, for which we devise an algorithm to efficiently retrieve communities. Specifically, we demonstrate that a conveniently parametrized form of regularized Laplacian matrix can be used to perform spectral clustering in sparse networks, without suffering from its degree heterogeneity. Besides, we exhibit important connections between this proposed matrix and the now popular non-backtracking matrix, the Bethe-Hessian matrix, as well as the standard Laplacian matrix. Interesti…
In this paper, we show how sparse or isoperimetric cuts of a probability density function relate to Cheeger cuts of its principal eigenfunction, for appropriate definitions of `sparse cut' and `principal eigenfunction'. We construct these appropriate definitions of sparse cut and principal eigenfunction in the probability density setting. Then, we prove Cheeger and Buser type inequalities similar to those for the normalized graph Laplacian of Alon-Milman. We demonstrate that no such inequalities hold for most prior definitions of sparse cut and principal eigenfunction. We apply this result to …
Baidu Research（百度研究院）；Arizona State University（亚利桑那州立大学）
Sparse subspace clustering methods with sparsity induced by $\ell^{0}$-norm, such as $\ell^{0}$-Sparse Subspace Clustering ($\ell^{0}$-SSC)~\citep{YangFJYH16-L0SSC-ijcv}, are demonstrated to be more effective than its $\ell^{1}$ counterpart such as Sparse Subspace Clustering (SSC)~\citep{ElhamifarV13}. However, the theoretical analysis of $\ell^{0}$-SSC is restricted to clean data that lie exactly in subspaces. Real data often suffer from noise and they may lie close to subspaces. In this paper, we show that an optimal solution to the optimization problem of noisy $\ell^{0}$-SSC achieves subsp…
University of Sheffield（谢菲尔德大学）；Department of Computer Science（计算机科学系）
We consider the problem of data clustering with unidentified feature quality and when a small amount of labelled data is provided. An unsupervised sparse clustering method can be employed in order to detect the subgroup of features necessary for clustering and a semi-supervised method can use the labelled data to create constraints and enhance the clustering solution. In this paper we propose a K-Means variant that employs these techniques. We show that the algorithm maintains the high performance of other semi-supervised algorithms and in addition preserves the ability to identify informative…
This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabelled input data points, which are assumed to be in a union of low-dimensional subspaces. We show that a modified version of SSC is \emph{provably effective} in correctly identifying the underlying subspaces, even with noisy data. This extends theoretical guarantee of this algorithm to more practical settings and provides justification to the success of SSC in a class of real applications.
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…
Sparse Subspace Clustering (SSC) is a state-of-the-art method for clustering high-dimensional data points lying in a union of low-dimensional subspaces. However, while $\ell_1$ optimization-based SSC algorithms suffer from high computational complexity, other variants of SSC, such as Orthogonal Matching Pursuit-based SSC (OMP-SSC), lose clustering accuracy in pursuit of improving time efficiency. In this letter, we propose a novel Active OMP-SSC, which improves clustering accuracy of OMP-SSC by adaptively updating data points and randomly dropping data points in the OMP process, while still en…
The Chinese University of Hong Kong（香港中文大学）；The Hong Kong Polytechnic University（香港理工大学）
In this paper, we propose a randomly projected convex clustering model for clustering a collection of $n$ high dimensional data points in $\mathbb{R}^d$ with $K$ hidden clusters. Compared to the convex clustering model for clustering original data with dimension $d$, we prove that, under some mild conditions, the perfect recovery of the cluster membership assignments of the convex clustering model, if exists, can be preserved by the randomly projected convex clustering model with embedding dimension $m = O(ε^{-2}\log(n))$, where $0 < ε< 1$ is some given parameter. We further prove that the emb…
The development of algorithms for unsupervised pattern recognition by nonlinear clustering is a notable problem in data science. Markov clustering (MCL) is a renowned algorithm that simulates stochastic flows on a network of sample similarities to detect the structural organization of clusters in the data, but it has never been generalized to deal with data nonlinearity. Minimum Curvilinearity (MC) is a principle that approximates nonlinear sample distances in the high-dimensional feature space by curvilinear distances, which are computed as transversal paths over their minimum spanning tree, …
INRIA - Sierra Project-team（INRIA - Sierra 项目组）；Département d’Informatique de l’Ecole Normale Supérieure (CNRS - ENS - INRIA)（巴黎高等师范学院计算机科学系（CNRS - ENS - INRIA））
Clustering high-dimensional data often requires some form of dimensionality reduction, where clustered variables are separated from "noise-looking" variables. We cast this problem as finding a low-dimensional projection of the data which is well-clustered. This yields a one-dimensional projection in the simplest situation with two clusters, and extends naturally to a multi-label scenario for more than two clusters. In this paper, (a) we first show that this joint clustering and dimension reduction formulation is equivalent to previously proposed discriminative clustering frameworks, thus leadi…
We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of…
Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportionally represented in each cluster. To this end, we develop variants of both normalized and unnormalized constrained SC and show that they help find fairer clusterings on both synthetic and real data. We also provide a rigorous theoretical analysis of our algorithms on a natural vari…
Feature selection is an important and challenging task in high dimensional clustering. For example, in genomics, there may only be a small number of genes that are differentially expressed, which are informative to the overall clustering structure. Existing feature selection methods, such as Sparse K-means, rarely tackle the problem of accounting features that can only separate a subset of clusters. In genomics, it is highly likely that a gene can only define one subtype against all the other subtypes or distinguish a pair of subtypes but not others. In this paper, we propose a K-means based c…
Functional neuroimaging can measure the brain?s response to an external stimulus. It is used to perform brain mapping: identifying from these observations the brain regions involved. This problem can be cast into a linear supervised learning task where the neuroimaging data are used as predictors for the stimulus. Brain mapping is then seen as a support recovery problem. On functional MRI (fMRI) data, this problem is particularly challenging as i) the number of samples is small due to limited acquisition time and ii) the variables are strongly correlated. We propose to overcome these difficult…
University of California, Berkeley（加州大学伯克利分校）；The Johns Hopkins University（约翰斯·霍普金斯大学）
In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representatio…
Sparse Subspace Clustering (SSC) is a popular unsupervised machine learning method for clustering data lying close to an unknown union of low-dimensional linear subspaces; a problem with numerous applications in pattern recognition and computer vision. Even though the behavior of SSC for complete data is by now well-understood, little is known about its theoretical properties when applied to data with missing entries. In this paper we give theoretical guarantees for SSC with incomplete data, and analytically establish that projecting the zero-filled data onto the observation pattern of the poi…
Sparse and geometry-aware generalisation of the mutual information for joint discriminative clustering and feature selection
Université Côte d’Azur（蔚蓝海岸大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Maasai Team（马赛团队）；CHU de Québec Research Centre（魁北克大学中心医院研究中心）；Laval University（拉瓦尔大学）
Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measurement, computation or privacy. In this paper, we study the theoretical properties of a popular subspace clustering algorithm named sparse subspace clustering (SSC) and establish formal success conditions of SSC on dimensionality-reduced data. Our analysis applies to the most general …
The Johns Hopkins University（约翰斯·霍普金斯大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
For a certain class of distributions, we prove that the linear programming relaxation of $k$-medoids clustering---a variant of $k$-means clustering where means are replaced by exemplars from within the dataset---distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large. Our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball; in this case, …
We address the problem of validating the ouput of clustering algorithms. Given data $\mathcal{D}$ and a partition $\mathcal{C}$ of these data into $K$ clusters, when can we say that the clusters obtained are correct or meaningful for the data? This paper introduces a paradigm in which a clustering $\mathcal{C}$ is considered meaningful if it is good with respect to a loss function such as the K-means distortion, and stable, i.e. the only good clustering up to small perturbations. Furthermore, we present a generic method to obtain post-inference guarantees of near-optimality and stability for a…
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 …
Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of constrained optimization problems with nonquadratic objective and constraints can be understood as nonlinear eigenproblems. We derive a generalization of the inverse power method which is guaranteed to converge to a nonlinear eigenvector. We apply the inverse power method to 1-spect…
Sparse subspace clustering (SSC) using greedy-based neighbor selection, such as matching pursuit (MP) and orthogonal matching pursuit (OMP), has been known as a popular computationally-efficient alternative to the conventional L1-minimization based methods. Under deterministic bounded noise corruption, in this paper we derive coherence-based sufficient conditions guaranteeing correct neighbor identification using MP/OMP. Our analyses exploit the maximum/minimum inner product between two noisy data points subject to a known upper bound on the noise level. The obtained sufficient condition clear…
University of Oxford（牛津大学）；Alan Turing Institute（艾伦·图灵研究所）
In semi-supervised learning for classification, it is assumed that every ground truth class of data is present in the small labelled dataset. Many real-world sparsely-labelled datasets are plausibly not of this type. It could easily be the case that some classes of data are found only in the unlabelled dataset -- perhaps the labelling process was biased -- so we do not have any labelled examples to train on for some classes. We call this learning regime $\textit{semi-unsupervised learning}$, an extreme case of semi-supervised learning, where some classes have no labelled exemplars in the train…
