Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to large sampling variance, the number of non-zeros is uneven among rows of the projection matrix which is generated in one trial, and more data information may be lost after dimension reduction. To break this bottleneck, based on random sampling without replacement in statistics, this paper builds a stable sparse subspace embedded matrix (S-SSE), in which non-zeros are uniformly distributed. It is proved that the S-SSE is stabler than the existing matrix, and it can maintain Euclidean distance between points well after dimension reduction. Our empirical studies corroborate our theoretical findings and demonstrate that our approach can indeed achieve satisfactory performance.
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have not been well studied. Our key contribution is to show that $2K$ projection vectors are sufficient for the independence preservation of any $K$ class data sampled from a union of independent subspaces. It is this non-trivial observation that we use for designing our dimensionality reduction technique. In this paper, we propose a novel dimensionality reductio…
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 …
Manifold learning is used for dimensionality reduction, with the goal of finding a projection subspace to increase and decrease the inter- and intraclass variances, respectively. However, a bottleneck for subspace learning methods often arises from the high dimensionality of datasets. In this paper, a hierarchical approach is proposed to scale subspace learning methods, with the goal of improving classification in large datasets by a range of 3% to 10%. Different combinations of methods are studied. We assess the proposed method on five publicly available large datasets, for different eigen-va…
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…
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched and the effect of random linear projections on measures such as intrinsic dimensionality still is an ongoing effort. In this paper, we investigate the less explored depths of linear projections onto explicit subspaces of varying dimensionality and the expectations of variance that ensue. The result is a new family of bounds for Euclidean distances and inner pr…
School of Computing, National University of Singapore（新加坡国立大学计算机学院）；Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）
Sensor drift is a well-known issue in the field of sensors and measurement and has plagued the sensor community for many years. In this paper, we propose a sensor drift correction method to deal with the sensor drift problem. Specifically, we propose a discriminative subspace projection approach for sensor drift reduction in electronic noses. The proposed method inherits the merits of the subspace projection method called domain regularized component analysis. Moreover, the proposed method takes the source data label information into consideration, which minimizes the within-class variance of …
In the machine learning field, dimensionality reduction is an important task. It mitigates the undesired properties of high-dimensional spaces to facilitate classification, compression, and visualization of high-dimensional data. During the last decade, researchers proposed many new (non-linear) techniques for dimensionality reduction. Most of these techniques are based on the intuition that data lies on or near a complex low-dimensional manifold that is embedded in the high-dimensional space. New techniques for dimensionality reduction aim at identifying and extracting the manifold from the h…
Technical University of Crete（克里特技术大学）；National Technical University of Athens（雅典国家技术大学）；University of West Attica（西阿提卡大学）
In this work we propose a method for reducing the dimensionality of tensor objects in a binary classification framework. The proposed Common Mode Patterns method takes into consideration the labels' information, and ensures that tensor objects that belong to different classes do not share common features after the reduction of their dimensionality. We experimentally validate the proposed supervised subspace learning technique and compared it against Multilinear Principal Component Analysis using a publicly available hyperspectral imaging dataset. Experimental results indicate that the proposed…
Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, whose number, orientations, and dimensions are all unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from undersampling due to complexity and speed constraints on the acquisition device or mechanism. More pertinently, even if the high-dimensional data set is available it is often desirable to first project the data points into a lower-dimensional space and to perform clustering there; th…
Huazhong University of Science and Technology（华中科技大学）；University of Technology Sydney（悉尼科技大学）
Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label inf…
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 …
Facebook（脸书）；Broad Institute of Harvard and MIT（哈佛大学与麻省理工学院博德研究所）
Bayesian optimization (BO) is a powerful paradigm for efficient optimization of black-box objective functions. High-dimensional BO presents a particular challenge, in part because the curse of dimensionality makes it difficult to define -- as well as do inference over -- a suitable class of surrogate models. We argue that Gaussian process surrogate models defined on sparse axis-aligned subspaces offer an attractive compromise between flexibility and parsimony. We demonstrate that our approach, which relies on Hamiltonian Monte Carlo for inference, can rapidly identify sparse subspaces relevant…
Outlying Aspect Mining (OAM) aims to find the subspaces (a.k.a. aspects) in which a given query is an outlier with respect to a given dataset. Existing OAM algorithms use traditional distance/density-based outlier scores to rank subspaces. Because these distance/density-based scores depend on the dimensionality of subspaces, they cannot be compared directly between subspaces of different dimensionality. $Z$-score normalisation has been used to make them comparable. It requires to compute outlier scores of all instances in each subspace. This adds significant computational overhead on top of al…
The University of Melbourne（墨尔本大学）；Tsinghua University（清华大学）；National Institute of Informatics（国立信息学研究所）
Datasets with significant proportions of noisy (incorrect) class labels present challenges for training accurate Deep Neural Networks (DNNs). We propose a new perspective for understanding DNN generalization for such datasets, by investigating the dimensionality of the deep representation subspace of training samples. We show that from a dimensionality perspective, DNNs exhibit quite distinctive learning styles when trained with clean labels versus when trained with a proportion of noisy labels. Based on this finding, we develop a new dimensionality-driven learning strategy, which monitors the…
Over the past few decades, we have witnessed a large family of algorithms that have been designed to provide different solutions to the problem of dimensionality reduction (DR). The DR is an essential tool to excavate the important information from the high-dimensional data by mapping the data to a low-dimensional subspace. Furthermore, for the diversity of varied high-dimensional data, the multi-view features can be utilized for improving the learning performance. However, many DR methods fail to integrating multiple views. Although the features from different views are extracted by different…
School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）；Department of Mathematics, Shandong Normal University（山东师范大学数学学院）
The implementation of conventional sparse principal component analysis (SPCA) on high-dimensional data sets has become a time consuming work. In this paper, a series of subspace projections are constructed efficiently by using Household QR factorization. With the aid of these subspace projections, a fast deflation method, called SPCA-SP, is developed for SPCA. This method keeps a good tradeoff between various criteria, including sparsity, orthogonality, explained variance, balance of sparsity, and computational cost. Comparative experiments on the benchmark data sets confirm the effectiveness …
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 …
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…
National Chung Hsing University（国立中兴大学）；IBM Research（IBM研究院）
Understanding and characterizing the subspaces of adversarial examples aid in studying the robustness of deep neural networks (DNNs) to adversarial perturbations. Very recently, Ma et al. (ICLR 2018) proposed to use local intrinsic dimensionality (LID) in layer-wise hidden representations of DNNs to study adversarial subspaces. It was demonstrated that LID can be used to characterize the adversarial subspaces associated with different attack methods, e.g., the Carlini and Wagner's (C&W) attack and the fast gradient sign attack. In this paper, we use MNIST and CIFAR-10 to conduct two new sets o…
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…
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…
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…
The Ohio State University（俄亥俄州立大学）；New York University（纽约大学）
Given labeled points in a high-dimensional vector space, we seek a low-dimensional subspace such that projecting onto this subspace maintains some prescribed distance between points of differing labels. Intended applications include compressive classification. Taking inspiration from large margin nearest neighbor classification, this paper introduces a semidefinite relaxation of this problem. Unlike its predecessors, this relaxation is amenable to theoretical analysis, allowing us to provably recover a planted projection operator from the data.
National University of Singapore（新加坡国立大学）；Arizona State University（亚利桑那州立大学）；Samsung Electronics（三星电子）；Georgia Institute of Technology（佐治亚理工学院）
Multivariate functional data from a complex system are naturally high-dimensional and have complex cross-correlation structure. The complexity of data structure can be observed as that (1) some functions are strongly correlated with similar features, while some others may have almost no cross-correlations with quite diverse features; and (2) the cross-correlation structure may also change over time due to the system evolution. With this regard, this paper presents a dynamic subspace learning method for multivariate functional data modeling. In particular, we consider different functions come f…
In this note we discuss a common misconception, namely that embeddings are always used to reduce the dimensionality of the item space. We show that when we measure dimensionality in terms of information entropy then the embedding of sparse probability distributions, that can be used to represent sparse features or data, may or not reduce the dimensionality of the item space. However, the embeddings do provide a different and often more meaningful representation of the items for a particular task at hand. Also, we give upper bounds and more precise guidelines for choosing the embedding dimensio…
Nanyang Technological University（南洋理工大学）；University of Vermont（佛蒙特大学）
It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. ME…
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 …
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…
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…
The sparse pseudo-input Gaussian process (SPGP) is a new approximation method for speeding up GP regression in the case of a large number of data points N. The approximation is controlled by the gradient optimization of a small set of M `pseudo-inputs', thereby reducing complexity from N^3 to NM^2. One limitation of the SPGP is that this optimization space becomes impractically big for high dimensional data sets. This paper addresses this limitation by performing automatic dimensionality reduction. A projection of the input space to a low dimensional space is learned in a supervised manner, al…
High-dimensional streaming data are becoming increasingly ubiquitous in many fields. They often lie in multiple low-dimensional subspaces, and the manifold structures may change abruptly on the time scale due to pattern shift or occurrence of anomalies. However, the problem of detecting the structural changes in a real-time manner has not been well studied. To fill this gap, we propose a dynamic sparse subspace learning approach for online structural change-point detection of high-dimensional streaming data. A novel multiple structural change-point model is proposed and the asymptotic properti…
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.…
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…
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…
Jiangnan University（江南大学）；The Hong Kong Polytechnic University（香港理工大学）
The superior interpretability and uncertainty modeling ability of Takagi-Sugeno-Kang fuzzy system (TSK FS) make it possible to describe complex nonlinear systems intuitively and efficiently. However, classical TSK FS usually adopts the whole feature space of the data for model construction, which can result in lengthy rules for high-dimensional data and lead to degeneration in interpretability. Furthermore, for highly nonlinear modeling task, it is usually necessary to use a large number of rules which further weakens the clarity and interpretability of TSK FS. To address these issues, a conci…
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 …
Most artificial networks today rely on dense representations, whereas biological networks rely on sparse representations. In this paper we show how sparse representations can be more robust to noise and interference, as long as the underlying dimensionality is sufficiently high. A key intuition that we develop is that the ratio of the operable volume around a sparse vector divided by the volume of the representational space decreases exponentially with dimensionality. We then analyze computationally efficient sparse networks containing both sparse weights and activations. Simulations on MNIST …
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…
Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the problem of whether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. It is now well-understood that such recovery is possible by practical algorithms when the dictionary $A$ is incoherent or restricted isometric. In this paper, we consider the more general case where $b$ lies in a subspace $\mathcal{S}_0$ spanned by a subset of linearly dependent columns of $A$,…
This work studies the robust subspace tracking (ST) problem. Robust ST can be simply understood as a (slow) time-varying subspace extension of robust PCA. It assumes that the true data lies in a low-dimensional subspace that is either fixed or changes slowly with time. The goal is to track the changing subspaces over time in the presence of additive sparse outliers and to do this quickly (with a short delay). We introduce a "fast" mini-batch robust ST solution that is provably correct under mild assumptions. Here "fast" means two things: (i) the subspace changes can be detected and the subspac…
The most effective dimensionality reduction procedures produce interpretable features from the raw input space while also providing good performance for downstream supervised learning tasks. For many methods, this requires optimizing one or more hyperparameters for a specific task, which can limit generalizability. In this study we propose sparse compressed agglomeration (SparCA), a novel dimensionality reduction procedure that involves a multistep hierarchical feature grouping, compression, and feature selection process. We demonstrate the characteristics and performance of the SparCA method …
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…
Is it possible to find the sparsest vector (direction) in a generic subspace $\mathcal{S} \subseteq \mathbb{R}^p$ with $\mathrm{dim}(\mathcal{S})= n < p$? This problem can be considered a homogeneous variant of the sparse recovery problem, and finds connections to sparse dictionary learning, sparse PCA, and many other problems in signal processing and machine learning. In this paper, we focus on a **planted sparse model** for the subspace: the target sparse vector is embedded in an otherwise random subspace. Simple convex heuristics for this planted recovery problem provably break down when th…
Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院（KIT））
In the real world, data streams are ubiquitous -- think of network traffic or sensor data. Mining patterns, e.g., outliers or clusters, from such data must take place in real time. This is challenging because (1) streams often have high dimensionality, and (2) the data characteristics may change over time. Existing approaches tend to focus on only one aspect, either high dimensionality or the specifics of the streaming setting. For static data, a common approach to deal with high dimensionality -- known as subspace search -- extracts low-dimensional, `interesting' projections (subspaces), in w…
Adform ApS（Adform公司）；Technical University of Denmark（丹麦技术大学）
In online advertising, display ads are increasingly being placed based on real-time auctions where the advertiser who wins gets to serve the ad. This is called real-time bidding (RTB). In RTB, auctions have very tight time constraints on the order of 100ms. Therefore mechanisms for bidding intelligently such as clickthrough rate prediction need to be sufficiently fast. In this work, we propose to use dimensionality reduction of the user-website interaction graph in order to produce simplified features of users and websites that can be used as predictors of clickthrough rate. We demonstrate tha…
The monitoring and management of high-volume feature-rich traffic in large networks offers significant challenges in storage, transmission and computational costs. The predominant approach to reducing these costs is based on performing a linear mapping of the data to a low-dimensional subspace such that a certain large percentage of the variance in the data is preserved in the low-dimensional representation. This variance-based subspace approach to dimensionality reduction forces a fixed choice of the number of dimensions, is not responsive to real-time shifts in observed traffic patterns, and…
This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves in…
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 …
Sparse Subspace Clustering (SSC) has achieved state-of-the-art clustering quality by performing spectral clustering over a $\ell^{1}$-norm based similarity graph. However, SSC is a transductive method which does not handle with the data not used to construct the graph (out-of-sample data). For each new datum, SSC requires solving $n$ optimization problems in O(n) variables for performing the algorithm over the whole data set, where $n$ is the number of data points. Therefore, it is inefficient to apply SSC in fast online clustering and scalable graphing. In this letter, we propose an inductive…
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 …
We consider the general dimensionality reduction problem of locating in a high-dimensional data cloud, a $k$-dimensional non-Gaussian subspace of interesting features. We use a projection pursuit approach -- we search for mutually orthogonal unit directions which maximise the 2-Wasserstein distance of the empirical distribution of data-projections along these directions from a standard Gaussian. Under a generative model, where there is a underlying (unknown) low-dimensional non-Gaussian subspace, we prove rigorous statistical guarantees on the accuracy of approximating this unknown subspace by…
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…
New York University（纽约大学）；University of Denver（丹佛大学）；Chinese University of Hong Kong（香港中文大学）；ShanghaiTech University（上海科技大学）；Columbia University（哥伦比亚大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning. However, in contrast to the classical sparse recovery problem, the most natural formulation for finding the sparsest vector in a subspace is usually nonconvex. In this paper, we overview recent advances on global nonconvex optimization theory for solving this problem, r…
University of Electronic Science and Technology of China（电子科技大学）；University of Alberta（阿尔伯塔大学）；University of Waterloo（滑铁卢大学）
As we aim at alleviating the curse of high-dimensionality, subspace learning is becoming more popular. Existing approaches use either information about global or local structure of the data, and few studies simultaneously focus on global and local structures as the both of them contain important information. In this paper, we propose a global and local structure preserving sparse subspace learning (GLoSS) model for unsupervised feature selection. The model can simultaneously realize feature selection and subspace learning. In addition, we develop a greedy algorithm to establish a generic combi…
