pairwise sparsity preserving embedding for unsupervised subspace learning and classification two novel unsupervised dimensionality reduction techniques termed sparse distance preserving embedding sdpe and sparse proximity preserving embedding sppe are proposed for feature extraction and classification sdpe and sppe perform in the clean data space recovered by sparse representation and enhanced euclidean distances over noise removed data are employed to measure pairwise similarities of points in extracting informative features sdpe and sppe aim at preserving pairwise similarities between data points in addition to preserving the sparse characteristics this paper calculates the sparsest representation of all vectors jointly by a convex optimization the sparsest codes enable certain local information of data to be preserved and can endow sdpe and sppe a natural discriminating power adaptive neighborhood and robust characteristic against noise and errors in delivering low dimensional embeddings we also mathematically show sdpe and sppe can be effectively extended for discriminant learning in a supervised manner the validity of sdpe and sppe is examined by extensive simulations comparison with other related state of the art unsupervised algorithms show that promising results are delivered by our techniques

information theoretic subspace clustering this paper addresses the problem of grouping the data points sampled from a union of multiple subspaces in the presence of outliers information theoretic objective functions are proposed to combine structured low rank representations lrrs to capture the global structure of data and information theoretic measures to handle outliers in theoretical part we point out that group sparsity induced measures ell 2 1 norm ell alpha norm and correntropy can be justified from the viewpoint of half quadratic hq optimization which facilitates both convergence study and algorithmic development in particular a general formulation is accordingly proposed to unify hq based group sparsity methods into a common framework in algorithmic part we develop information theoretic subspace clustering methods via correntropy with the help of parzen window estimation correntropy is used to handle either outliers under any distributions or sample specific errors in data pairwise link constraints are further treated as a prior structure of lrrs based on the hq framework iterative algorithms are developed to solve the nonconvex information theoretic loss functions experimental results on three benchmark databases show that our methods can further improve the robustness of lrr subspace clustering and outperform other state of the art subspace clustering methods

probabilistic class structure regularized sparse representation graph for semi supervised hyperspectral image classification abstract graph based semi supervised learning ssl which performs well in hyperspectral image classification with a small amount of labeled samples has drawn a lot of attention in the past few years the key step of graph based ssl is to construct a good graph to represent original data structures among the existing graph construction methods sparse representation sr based methods have shown impressive performance on graph based ssl however most sr based methods fail to take into consideration the class structure of data in ssl we can obtain a probabilistic class structure which implies the probabilistic relationship between each sample and each class of the whole data by utilizing a small amount of labeled samples such class structure information can help sr model to yield a more discriminative coefficients which motivates us to exploit this class structure information in order to learn a discriminative graph in this paper we present a discriminative graph construction method called probabilistic class structure regularized sparse representation pcssr approach by incorporating the class structure information into the sr model pcssr can learn a discriminative graph from the data a class structure regularization is developed to make use of the probabilistic class structure and therefore to improve the discriminability of the graph we formulate our problem as a constrained sparsity minimization problem and solve it by the alternating direction method with adaptive penalty admap the experimental results on hyperion and aviris hyperspectral data show that our method outperforms state of the art

vector sparse representation of color image using quaternion matrix analysis traditional sparse image models treat color image pixel as a scalar which represents color channels separately or concatenate color channels as a monochrome image in this paper we propose a vector sparse representation model for color images using quaternion matrix analysis as a new tool for color image representation its potential applications in several image processing tasks are presented including color image reconstruction denoising inpainting and super resolution the proposed model represents the color image as a quaternion matrix where a quaternion based dictionary learning algorithm is presented using the k quaternion singular value decomposition qsvd generalized k means clustering for qsvd method it conducts the sparse basis selection in quaternion space which uniformly transforms the channel images to an orthogonal color space in this new color space it is significant that the inherent color structures can be completely preserved during vector reconstruction moreover the proposed sparse model is more efficient comparing with the current sparse models for image restoration tasks due to lower redundancy between the atoms of different color channels the experimental results demonstrate that the proposed sparse image model avoids the hue bias issue successfully and shows its potential as a general and powerful tool in color image analysis and processing domain

multibody grouping from motion images we want to deduce from a sequence of noisy two dimensional images of a scene of several rigid bodies moving independently in three dimensions the number of bodies and the grouping of given feature points in the images to the bodies prior processing is assumed to have identified features or points common to all frames and the images are assumed to be created by orthographic projection i e perspective effects are minimal we describe a computationally inexpensive algorithm that can determine which points or features belong to which rigid body using the fact that with exact observations in orthographic projection points on a single body lie in a three or less dimensional linear manifold of frame space if there are enough observations and independent motions these manifolds can be viewed as a set linearly independent four or less dimensional subspaces we show that the row echelon canonical form provides direct information on the grouping of points to these subspaces treatment of the noise is the most difficult part of the problem this paper uses a statistical approach to estimate the grouping of points to subspaces in the presence of noise by computing which partition has the maximum likelihood the input data is assumed to be contaminated with independent gaussian noise the algorithm can base its estimates on a user supplied standard deviation of the noise or it can estimate the noise from the data the algorithm can also be used to estimate the probability of a user specified partition so that the hypothesis can be combined with others using bayesian statistics

robust algebraic segmentation of mixed rigid body and planar motions from two views this paper studies segmentation of multiple rigid body motions in a 3 d dynamic scene under perspective camera projection we consider dynamic scenes that contain both 3 d rigid body structures and 2 d planar structures based on the well known epipolar and homography constraints between two views we propose a hybrid perspective constraint hpc to unify the representation of rigid body and planar motions given a mixture of k hybrid perspective constraints we propose an algebraic process to partition image correspondences to the individual 3 d motions called robust algebraic segmentation ras particularly we prove that the joint distribution of image correspondences is uniquely determined by a set of 2k th degree polynomials a global signature for the union of k motions of possibly mixed type the first and second derivatives of these polynomials provide a means to recover the association of the individual image samples to their respective motions finally using robust statistics we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers we conduct extensive simulations and real experiments to validate the performance of the new algorithm the results demonstrate that ras achieves notably higher accuracy than most existing robust motion segmentation methods including random sample consensus ransac and its variations the implementation of the algorithm is also two to three times faster than the existing methods the implementation of the algorithm and the benchmark scripts are available at http perception csl illinois edu ras

unsupervised object segmentation with a hybrid graph model hgm in this work we address the problem of performing class specific unsupervised object segmentation i e automatic segmentation without annotated training images object segmentation can be regarded as a special data clustering problem where both class specific information and local texture color similarities have to be considered to this end we propose a hybrid graph model hgm that can make effective use of both symmetric and asymmetric relationship among samples the vertices of a hybrid graph represent the samples and are connected by directed edges and or undirected ones which represent the asymmetric and or symmetric relationship between them respectively when applied to object segmentation vertices are superpixels the asymmetric relationship is the conditional dependence of occurrence and the symmetric relationship is the color texture similarity by combining the markov chain formed by the directed subgraph and the minimal cut of the undirected subgraph the object boundaries can be determined for each image using the hgm we can conveniently achieve simultaneous segmentation and recognition by integrating both top down and bottom up information into a unified process experiments on 42 object classes 9 415 images in total show promising results

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

robust regression discriminative methods e g kernel regression svm have been extensively used to solve problems such as object recognition image alignment and pose estimation from images these methods typically map image features mathbf x to continuous e g pose or discrete e g object category values a major drawback of existing discriminative methods is that samples are directly projected onto a subspace and hence fail to account for outliers common in realistic training sets due to occlusion specular reflections or noise it is important to notice that existing discriminative approaches assume the input variables mathbf x to be noise free thus discriminative methods experience significant performance degradation when gross outliers are present despite its obvious importance the problem of robust discriminative learning has been relatively unexplored in computer vision this paper develops the theory of robust regression rr and presents an effective convex approach that uses recent advances on rank minimization the framework applies to a variety of problems in computer vision including robust linear discriminant analysis regression with missing data and multi label classification several synthetic and real examples with applications to head pose estimation from images image and video classification and facial attribute classification with missing data are used to illustrate the benefits of rr

graph regularized restricted boltzmann machine the restricted boltzmann machine rbm has received an increasing amount of interest in recent years it determines good mapping weights that capture useful latent features in an unsupervised manner the rbm and its generalizations have been successfully applied to a variety of image classification and speech recognition tasks however most of the existing rbm based models disregard the preservation of the data manifold structure in many real applications the data generally reside on a low dimensional manifold embedded in high dimensional ambient space in this brief we propose a novel graph regularized rbm to capture features and learning representations explicitly considering the local manifold structure of the data by imposing manifold based locality that preserves constraints on the hidden layer of the rbm the model ultimately learns sparse and discriminative representations the representations can reflect data distributions while simultaneously preserving the local manifold structure of data we test our model using several benchmark image data sets for unsupervised clustering and supervised classification problem the results demonstrate that the performance of our method exceeds the state of the art alternatives

a regularized approach for geodesic based semisupervised multimanifold learning geodesic distance as an essential measurement for data dissimilarity has been successfully used in manifold learning however most geodesic distance based manifold learning algorithms have two limitations when applied to classification 1 class information is rarely used in computing the geodesic distances between data points on manifolds and 2 little attention has been paid to building an explicit dimension reduction mapping for extracting the discriminative information hidden in the geodesic distances in this paper we regard geodesic distance as a kind of kernel which maps data from linearly inseparable space to linear separable distance space in doing this a new semisupervised manifold learning algorithm namely regularized geodesic feature learning algorithm is proposed the method consists of three techniques a semisupervised graph construction method replacement of original data points with feature vectors which are built by geodesic distances and a new semisupervised dimension reduction method for feature vectors experiments on the mnist usps handwritten digit data sets mit cbcl face versus nonface data set and an intelligent traffic data set show the effectiveness of the proposed algorithm

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

relations among some low rank subspace recovery models recovering intrinsic low dimensional subspaces from data distributed on them is a key preprocessing step to many applications in recent years a lot of work has modeled subspace recovery as low rank minimization problems we find that some representative models such as robust principal component analysis r pca robust low rank representation r lrr and robust latent low rank representation r latlrr are actually deeply connected more specifically we discover that once a solution to one of the models is obtained we can obtain the solutions to other models in closed form formulations since r pca is the simplest our discovery makes it the center of low rank subspace recovery models our work has two important implications first r pca has a solid theoretical foundation under certain conditions we could find globally optimal solutions to these low rank models at an overwhelming probability although these models are nonconvex second we can obtain significantly faster algorithms for these models by solving r pca first the computation cost can be further cut by applying low complexity randomized algorithms for example our novel filtering algorithm to r pca although for the moment the formal proof of our filtering algorithm is not yet available experiments verify the advantages of our algorithm over other state of the art methods based on the alternating direction method

self regularized fixed rank representation for subspace segmentation abstract consider a set of data points generated from various linear subspaces subspace segmentation tasks which are important in fields such as computer vision and image processing aim to partition the set of data points so as to recover these subspaces the subspace segmentation method fixed rank representation frr was introduced to remedy the problem of insufficient sampling in classical low rank representation lrr in many subspace segmentation applications frr has achieved much better results than those of lrr in this paper a new frr related algorithm called self regularized fixed rank representation srfrr is proposed in srfrr a laplacian regularizer is constructed using the coefficient matrix obtained by srfrr itself further by proving that the laplacian regularizer can be transformed into a structure constraint on the coefficient matrix we show that another existing method sparse frr sfrr is a special case of srfrr to implement the srfrr method we present two optimization algorithms experiments on both synthetic and real databases show that srfrr outperforms some existing frr and lrr related algorithms

subspace learning by l0 induced sparsity subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure such methods with sparsity prior such as sparse subspace clustering ssc elhamifar and vidal in ieee trans pattern anal mach intell 35 11 2765 2781 2013 with the sparsity induced by the ell 1 norm are demonstrated to be effective in subspace clustering most of those methods require certain assumptions e g independence or disjointness on the subspaces however these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods in this paper we propose ell 0 induced sparse subspace clustering ell 0 ssc in contrast to the required assumptions such as independence or disjointness on subspaces for most existing sparse subspace clustering methods we prove that ell 0 ssc guarantees the subspace sparse representation a key element in subspace clustering for arbitrary distinct underlying subspaces almost surely under the mild i i d assumption on the data generation we also present the no free lunch theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding ell 0 sparse representation problem of ell 0 ssc a novel approximate algorithm named approximate ell 0 ssc a ell 0 ssc is developed which employs proximal gradient descent to obtain a sub optimal solution to the optimization problem of ell 0 ssc with theoretical guarantee the sub optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results extensive experimental results on various data sets demonstrate the superiority of a ell 0 ssc compared to other competing clustering methods furthermore we extend ell 0 ssc to semi supervised learning by performing label propagation on the sparse similarity matrix learnt by a ell 0 ssc and demonstrate the effectiveness of the resultant semi supervised learning method termed ell 0 sparse subspace label propagation ell 0 sslp

minimum error entropy based sparse representation for robust subspace clustering this paper addresses the problem of clustering data points that are approximately drawn from multiple subspaces recently a large family of spectral clustering based methods such as sparse subspace clustering ssc and low rank representation lrr have been proposed in this paper we present a general formulation to unify many of them within a common framework based on atomic representation since mean square error mse relies heavily on the gaussianity assumption the previous mse based subspace clustering methods have the limitation of being sensitive to non gaussian noise in this paper we develop a novel subspace clustering method termed meessc by specifying the minimum error entropy mee as the loss function and the sparsity inducing atomic set we show that meessc can well overcome the above limitation the experimental results on both synthetic and real data verify the effectiveness of the proposed method

label information guided graph construction for semi supervised learning in the literature most existing graph based semi supervised learning methods only use the label information of observed samples in the label propagation stage while ignoring such valuable information when learning the graph in this paper we argue that it is beneficial to consider the label information in the graph learning stage specifically by enforcing the weight of edges between labeled samples of different classes to be zero we explicitly incorporate the label information into the state of the art graph learning methods such as the low rank representation lrr and propose a novel semi supervised graph learning method called semi supervised low rank representation this results in a convex optimization problem with linear constraints which can be solved by the linearized alternating direction method though we take lrr as an example our proposed method is in fact very general and can be applied to any self representation graph learning methods experiment results on both synthetic and real data sets demonstrate that the proposed graph learning method can better capture the global geometric structure of the data and therefore is more effective for semi supervised learning tasks

a simple prior free method for non rigid structure from motion factorization this paper proposes a simple prior free method for solving the non rigid structure from motion nrsfm factorization problem other than using the fundamental low order linear combination model assumption our method does not assume any extra prior knowledge either about the non rigid structure or about the camera motions yet it works effectively and reliably producing optimal results and not suffering from the inherent basis ambiguity issue which plagued most conventional nrsfm factorization methods our method is very simple to implement which involves solving a very small sdp semi definite programming of fixed size and a nuclear norm minimization problem we also present theoretical analysis on the uniqueness and the relaxation gap of our solutions extensive experiments on both synthetic and real motion capture data assuming following the low order linear combination model are conducted which demonstrate that our method indeed outperforms most of the existing non rigid factorization methods this work offers not only new theoretical insight but also a practical everyday solution to nrsfm

motion segmentation in the presence of outlying incomplete or corrupted trajectories in this paper we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence using the affine camera model this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces in practice due to limitations of the tracker occlusions and the presence of nonrigid objects in the scene the obtained motion trajectories may contain grossly mistracked features missing entries or corrupted entries in this paper we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework our methods draw strong connections between lossy compression rank minimization and sparse representation we test our methods extensively on the hopkins155 motion segmentation database and other motion sequences with outliers and missing data we compare the performance of our methods to state of the art motion segmentation methods based on expectation maximization and spectral clustering for data without outliers or missing information the results of our methods are on par with the state of the art results and in many cases exceed them in addition our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above all code and results are publicly available at http perception csl uiuc edu coding motion

robust recovery of multiple subspaces by geometric lp minimization we assume i i d data sampled from a mixture distribution with k components along fixed d dimensional linear subspaces and an additional outlier component for p 0 we study the simultaneous recovery of the k fixed subspaces by minimizing the lp averaged distances of the sampled data points from any k subspaces under some conditions we show that if 0 1 and p 1 then the underlying subspaces cannot be recovered or even nearly recovered by lp minimization the results of this paper partially explain the successes and failures of the basic approach of lp energy minimization for modeling data by multiple subspaces

low rank matrix completion in the presence of high coherence prevalent matrix completion methods capture only the low rank property which gives merely a constraint that the data points lie on some low dimensional subspace but generally ignore the extra structures beyond low rank that specify in more detail how the data points lie on the subspace whenever the data points are not uniformly distributed on the low dimensional subspace the row coherence of the target matrix to recover could be considerably high and accordingly prevalent methods might fail even if the target matrix is fairly low rank to relieve this challenge we suggest to consider a model termed low rank factor decomposition lrfd which imposes an additional restriction that the data points must be represented as linear compressive combinations of the bases in a given dictionary we show that lrfd can effectively mitigate the challenges of high row coherence provided that its dictionary is configured properly namely it is mathematically proven that if the dictionary is well conditioned and low rank then lrfd can weaken the dependence on the row coherence in particular if the dictionary itself is low rank then the dependence on the row coherence can be entirely removed subsequently we devise two practical algorithms to obtain proper dictionaries in unsupervised environments one uses the existing matrix completion methods to construct the dictionary in lrfd and the other tries to learn a proper dictionary from the data given experiments on randomly generated matrices and motion datasets show superior performance of our proposed algorithms

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

active subspace toward scalable low rank learning we address the scalability issues in low rank matrix learning problems usually these problems resort to solving nuclear norm regularized optimization problems nnrops which often suffer from high computational complexities if based on existing solvers especially in large scale settings based on the fact that the optimal solution matrix to an nnrop is often low rank we revisit the classic mechanism of low rank matrix factorization based on which we present an active subspace algorithm for efficiently solving nnrops by transforming large scale nnrops into small scale problems the transformation is achieved by factorizing the large solution matrix into the product of a small orthonormal matrix active subspace and another small matrix although such a transformation generally leads to nonconvex problems we show that a suboptimal solution can be found by the augmented lagrange alternating direction method for the robust pca rpca candes li ma wright 2009 problem a typical example of nnrops theoretical results verify the suboptimality of the solution produced by our algorithm for the general nnrops we empirically show that our algorithm significantly reduces the computational complexity without loss of optimality

low rank structure learning via nonconvex heuristic recovery in this paper we propose a nonconvex framework to learn the essential low rank structure from corrupted data different from traditional approaches which directly utilizes convex norms to measure the sparseness our method introduces more reasonable nonconvex measurements to enhance the sparsity in both the intrinsic low rank structure and the sparse corruptions we will respectively introduce how to combine the widely used lp norm 0 p 1 and log sum term into the framework of low rank structure learning although the proposed optimization is no longer convex it still can be effectively solved by a majorization minimization mm type algorithm with which the nonconvex objective function is iteratively replaced by its convex surrogate and the nonconvex problem finally falls into the general framework of reweighed approaches we prove that the mm type algorithm can converge to a stationary point after successive iterations the proposed model is applied to solve two typical problems robust principal component analysis and low rank representation experimental results on low rank structure learning demonstrate that our nonconvex heuristic methods especially the log sum heuristic recovery algorithm generally perform much better than the convex norm based method 0 p 1 for both data with higher rank and with denser corruptions

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

enhancing low rank subspace clustering by manifold regularization recently low rank representation lrr method has achieved great success in subspace clustering sc which aims to cluster the data points that lie in a union of low dimensional subspace given a set of data points lrr seeks the lowest rank representation among the many possible linear combinations of the bases in a given dictionary or in terms of the data itself however lrr only considers the global euclidean structure while the local manifold structure which is often important for many real applications is ignored in this paper to exploit the local manifold structure of the data a manifold regularization characterized by a laplacian graph has been incorporated into lrr leading to our proposed laplacian regularized lrr laplrr an efficient optimization procedure which is based on alternating direction method of multipliers admm is developed for laplrr experimental results on synthetic and real data sets are presented to demonstrate that the performance of lrr has been enhanced by using the manifold regularization

subspace clustering using a low rank constrained autoencoder the performance of subspace clustering is affected by data representation data representation for subspace clustering maps data from the original space into another space with the property of better separability many data representation methods have been developed in recent years typical among them are low rank representation lrr and an autoencoder lrr is a linear representation method that captures the global structure of data with low rank constraint alternatively an autoencoder nonlinearly maps data into a latent space using a neural network by minimizing the difference between the reconstruction and input to combine the advantages of an lrr globality and autoencoder self supervision based locality we propose a novel data representation method for subspace clustering the proposed method called low rank constrained autoencoder lrae forces the latent representation of the neural network to be of low rank and the low rank constraint is computed as a prior from the input space one major advantage of the lrae is that the learned data representation not only maintains the local features of the data but also preserves the underlying low rank global structure extensive experiments on several datasets for subspace clustering were conducted they demonstrated that the proposed lrae substantially outperformed state of the art subspace clustering methods

mrm lasso a sparse multiview feature selection method via low rank analysis learning about multiview data involves many applications such as video understanding image classification and social media however when the data dimension increases dramatically it is important but very challenging to remove redundant features in multiview feature selection in this paper we propose a novel feature selection algorithm multiview rank minimization based lasso mrm lasso which jointly utilizes lasso for sparse feature selection and rank minimization for learning relevant patterns across views instead of simply integrating multiple lasso from view level we focus on the performance of sample level sample significance and introduce pattern specific weights into mrm lasso the weights are utilized to measure the contribution of each sample to the labels in the current view in addition the latent correlation across different views is successfully captured by learning a low rank matrix consisting of pattern specific weights the alternating direction method of multipliers is applied to optimize the proposed mrm lasso experiments on four real life data sets show that features selected by mrm lasso have better multiview classification performance than the baselines moreover pattern specific weights are demonstrated to be significant for learning about multiview data compared with view specific weights

transformation invariant subspace clustering subspace clustering has achieved great success in many computer vision applications however most subspace clustering algorithms require well aligned data samples which is often not straightforward to achieve this paper proposes a transformation invariant subspace clustering framework by jointly aligning data samples and learning subspace representation by alignment the transformed data samples become highly correlated and a better affinity matrix can be obtained the joint problem can be reduced to a sequence of least squares regression problems which can be efficiently solved we verify the effectiveness of the proposed method with extensive experiments on unaligned real data demonstrating its higher clustering accuracy than the state of the art subspace clustering and transformation invariant clustering algorithms highlightsa transformation invariant subspace clustering algorithm is proposed alignment and subspace clustering are mutually dependent state of the art clustering results on the unconstrained face dataset

exploiting unsupervised and supervised constraints for subspace clustering data in many image and video analysis tasks can be viewed as points drawn from multiple low dimensional subspaces with each subspace corresponding to one category or class one basic task for processing such kind of data is to separate the points according to the underlying subspace referred to as subspace clustering extensive studies have been made on this subject and nearly all of them use unconstrained subspace models meaning the points can be drawn from everywhere of a subspace to represent the data in this paper we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces e g belonging to a submanifold or satisfying the spatial regularity constraint this assumption usually describes the real data better such as differently moving objects in a video scene and face images of different subjects under varying illumination a unified integer linear programming optimization framework is used to approach subspace clustering which can be efficiently solved by a branch and bound bb method we also show that various kinds of supervised information such as subspace number outlier ratio pairwise constraints size prior and etc can be conveniently incorporated into the proposed framework experiments on real data show that the proposed method outperforms the state of the art algorithms significantly in clustering accuracy the effectiveness of the proposed method in exploiting supervised information is also demonstrated

feature learning via partial differential equation with applications to face recognition a novel partial differential equation method is proposed for feature learning the feature is discriminative and invariant under rotation translation and illumination the method is efficient in low resolution images and when the samples are few to our best knowledge this is the first work that applies pde to feature learning feature learning is a critical step in pattern recognition such as image classification however most of the existing methods cannot extract features that are discriminative and at the same time invariant under some transforms this limits the classification performance especially in the case of small training sets to address this issue in this paper we propose a novel partial differential equation pde based method for feature learning the feature learned by our pde is discriminative also translationally and rotationally invariant and robust to illumination variation to our best knowledge this is the first work that applies pde to feature learning and image recognition tasks specifically we model feature learning as an evolution process governed by a pde which is designed to be translationally and rotationally invariant and is learned via minimizing the training error hence extracts discriminative information from data after feature extraction we apply a linear classifier for classification we also propose an efficient algorithm that optimizes the whole framework our method is very effective when the training samples are few the experimental results of face recognition on the four benchmark face datasets show that the proposed method outperforms the state of the art feature learning methods in the case of low resolution images and when the training samples are limited

convex sparse spectral clustering single view to multi view spectral clustering sc is one of the most widely used methods for data clustering it first finds a low dimensional embedding u of data by computing the eigenvectors of the normalized laplacian matrix and then performs k means on text u top to get the final clustering result in this paper we observe that in the ideal case text u text u top should be block diagonal and thus sparse therefore we propose the sparse sc ssc method that extends the sc with sparse regularization on text u text u top to address the computational issue of the nonconvex ssc model we propose a novel convex relaxation of ssc based on the convex hull of the fixed rank projection matrices then the convex ssc model can be efficiently solved by the alternating direction method of multipliers furthermore we propose the pairwise ssc that extends ssc to boost the clustering performance by using the multi view information of data experimental comparisons with several baselines on real world datasets testify to the efficacy of our proposed methods

double nuclear norm based matrix decomposition for occluded image recovery and background modeling robust principal component analysis rpca is a new emerging method for exact recovery of corrupted low rank matrices it assumes that the real data matrix has low rank and the error matrix is sparse this paper presents a method called double nuclear norm based matrix decomposition dnmd for dealing with the image data corrupted by continuous occlusion the method uses a unified low rank assumption to characterize the real image data and continuous occlusion specifically we assume all image vectors form a low rank matrix and each occlusion induced error image is a low rank matrix as well compared with rpca the low rank assumption of dnmd is more intuitive for describing occlusion moreover dnmd is solved by alternating direction method of multipliers our algorithm involves only one operator the singular value shrinkage operator dnmd as a transductive method is further extended into inductive dnmd idnmd both dnmd and idnmd use nuclear norm for measuring the continuous occlusion induced error while many previous methods use l 1 l 2 or other m estimators extensive experiments on removing occlusion from face images and background modeling from surveillance videos demonstrate the effectiveness of the proposed methods

joint low rank and sparse principal feature coding for enhanced robust representation and visual classification recovering low rank and sparse subspaces jointly for enhanced robust representation and classification is discussed technically we first propose a transductive low rank and sparse principal feature coding lspfc formulation that decomposes given data into a component part that encodes low rank sparse principal features and a noise fitting error part to well handle the outside data we then present an inductive lspfc i lspfc i lspfc incorporates embedded low rank and sparse principal features by a projection into one problem for direct minimization so that the projection can effectively map both inside and outside data into the underlying subspaces to learn more powerful and informative features for representation to ensure that the learned features by i lspfc are optimal for classification we further combine the classification error with the feature coding error to form a unified model discriminative lspfc d lspfc to boost performance the model of d lspfc seamlessly integrates feature coding and discriminative classification so the representation and classification powers can be enhanced the proposed approaches are more general and several recent existing low rank or sparse coding algorithms can be embedded into our problems as special cases visual and numerical results demonstrate the effectiveness of our methods for representation and classification

efficient recovery of low rank matrix via double nonconvex nonsmooth rank minimization recently there is a rapidly increasing attraction for the efficient recovery of low rank matrix in computer vision and machine learning the popular convex solution of rank minimization is nuclear norm based minimization nnm which usually leads to a biased solution since nnm tends to overshrink the rank components and treats each rank component equally to address this issue some nonconvex nonsmooth rank nnr relaxations have been exploited widely different from these convex and nonconvex rank substitutes this paper first introduces a general and flexible rank relaxation function named weighted nnr relaxation function which is actually derived from the initial double nnr dnnr relaxations i e dnnr relaxation function acts on the nonconvex singular values function svf an iteratively reweighted svf optimization algorithm with continuation technology through computing the supergradient values to define the weighting vector is devised to solve the dnnr minimization problem and the closed form solution of the subproblem can be efficiently obtained by a general proximal operator in which each element of the desired weighting vector usually satisfies the nondecreasing order we next prove that the objective function values decrease monotonically and any limit point of the generated subsequence is a critical point combining the kurdyka ojasiewicz property with some milder assumptions we further give its global convergence guarantee as an application in the matrix completion problem experimental results on both synthetic data and real world data can show that our methods are competitive with several state of the art convex and nonconvex matrix completion methods

robust face recognition based on dynamic rank representation abstract robust face recognition is an active topic in computer vision while face occlusion is one of the most challenging problems for robust face recognition algorithm the latest research on low rank representation demonstrated its high efficiency to subspace segmentation and feature extraction motivated by previous work in this paper we consider the problem of human face recognition from frontal views with varying illumination as well as occlusion and disguise we present a novel approach for face recognition by extracting dynamic subspace of images and obtaining the discriminative parts in each individual we use these parts to represent the characteristic of discriminative components give a recognition protocol to classify face images the experiments carried on publicly available databases i e ar extended yale b and orl vilidate its accuray robustness and speed the proposed method needs lower dimensions training samples but gains a higher recognition rate than other popular approaches

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

l 2 bounded variation and absolutely continuous spectrum of jacobi matrices we disprove a conjecture of breuer last simon breuer et al in constr approx 32 2 221 254 2010 concerning the absolutely continuous spectrum of jacobi matrices with coefficients that obey an l2 bounded variation condition with step q we prove existence of a c spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal

laplacian regularized low rank representation and its applications low rank representation lrr has recently attracted a great deal of attention due to its pleasing efficacy in exploring low dimensional subspace structures embedded in data for a given set of observed data corrupted with sparse errors lrr aims at learning a lowest rank representation of all data jointly lrr has broad applications in pattern recognition computer vision and signal processing in the real world data often reside on low dimensional manifolds embedded in a high dimensional ambient space however the lrr method does not take into account the non linear geometric structures within data thus the locality and similarity information among data may be missing in the learning process to improve lrr in this regard we propose a general laplacian regularized low rank representation framework for data representation where a hypergraph laplacian regularizer can be readily introduced into i e a non negative sparse hyper laplacian regularized lrr model nshlrr by taking advantage of the graph regularizer our proposed method not only can represent the global low dimensional structures but also capture the intrinsic non linear geometric information in data the extensive experimental results on image clustering semi supervised image classification and dimensionality reduction tasks demonstrate the effectiveness of the proposed method

blessing of dimensionality recovering mixture data via dictionary pursuit this paper studies the problem of recovering the authentic samples that lie on a union of multiple subspaces from their corrupted observations due to the high dimensional and massive nature of today s data driven community it is arguable that the target matrix i e authentic sample matrix to recover is often low rank in this case the recently established robust principal component analysis rpca method already provides us a convenient way to solve the problem of recovering mixture data however in general rpca is not good enough because the incoherent condition assumed by rpca is not so consistent with the mixture structure of multiple subspaces namely when the subspace number grows the row coherence of data keeps heightening and accordingly rpca degrades to overcome the challenges arising from mixture data we suggest to consider lrr in this paper we elucidate that lrr can well handle mixture data as long as its dictionary is configured appropriately more precisely we mathematically prove that lrr can weaken the dependence on the row coherence provided that the dictionary is well conditioned and has a rank of not too high in particular if the dictionary itself is sufficiently low rank then the dependence on the row coherence can be completely removed these provide some elementary principles for dictionary learning and naturally lead to a practical algorithm for recovering mixture data our experiments on randomly generated matrices and real motion sequences show promising results

robust volume minimization based matrix factorization for remote sensing and document clustering this paper considers volume minimization volmin based structured matrix factorization volmin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum volume simplex that encloses all the columns of the data matrix recent work showed that volmin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications this paper focuses on both theoretical and practical aspects of volmin on the theory side exact equivalence of two independently developed sufficient conditions for volmin identifiability is proven here thereby providing a more comprehensive understanding of this aspect of volmin on the algorithm side computational complexity and sensitivity to outliers are two key challenges associated with real world applications of volmin these are addressed here via a new volmin algorithm that handles volume regularization in a computationally simple way and automatically detects and iteratively downweights outliers simultaneously simulations and real data experiments using a remotely sensed hyperspectral image and the reuters document corpus are employed to showcase the effectiveness of the proposed algorithm

robust kernel low rank representation recently low rank representation lrr has shown promising performance in many real world applications such as face clustering however lrr may not achieve satisfactory results when dealing with the data from nonlinear subspaces since it is originally designed to handle the data from linear subspaces in the input space meanwhile the kernel based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel induced mapping to effectively cope with the nonlinear data we first propose the kernelized version of lrr in the clean data case we also present a closed form solution for the resultant optimization problem moreover to handle corrupted data we propose the robust kernel lrr rklrr approach and develop an efficient optimization algorithm to solve it based on the alternating direction method in particular we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved and it is guaranteed to obtain a globally optimal solution besides our proposed algorithm can also solve the original lrr problem which is a special case of our rklrr when using the linear kernel in addition based on our new optimization technique the kernelization of some variants of lrr can be similarly achieved comprehensive experiments on synthetic data sets and real world data sets clearly demonstrate the efficiency of our algorithm as well as the effectiveness of rklrr and the kernelization of two variants of lrr

spectral curvature clustering scc this paper presents novel techniques for improving the performance of a multi way spectral clustering framework govindu in proceedings of the 2005 ieee computer society conference on computer vision and pattern recognition cvpr 05 vol 1 pp 1150 1157 2005 chen and lerman 2007 preprint in the supplementary webpage for segmenting affine subspaces specifically it suggests an iterative sampling procedure to improve the uniform sampling strategy an automatic scheme of inferring the tuning parameter from the data a precise initialization procedure for k means as well as a simple strategy for isolating outliers the resulting algorithm spectral curvature clustering scc requires only linear storage and takes linear running time in the size of the data it is supported by theory which both justifies its successful performance and guides our practical choices we compare it with other existing methods on a few artificial instances of affine subspaces application of the algorithm to several real world problems is also discussed

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

robust transfer metric learning for image classification metric learning has attracted increasing attention due to its critical role in image analysis and classification conventional metric learning always assumes that the training and test data are sampled from the same or similar distribution however to build an effective distance metric we need abundant supervised knowledge i e side label information which is generally inaccessible in practice because of the expensive labeling cost in this paper we develop a robust transfer metric learning rtml framework to effectively assist the unlabeled target learning by transferring the knowledge from the well labeled source domain specifically rtml exploits knowledge transfer to mitigate the domain shift in two directions i e sample space and feature space in the sample space domain wise and class wise adaption schemes are adopted to bridge the gap of marginal and conditional distribution disparities across two domains in the feature space our metric is built in a marginalized denoising fashion and low rank constraint which make it more robust to tackle noisy data in reality furthermore we design an explicit rank constraint regularizer to replace the rank minimization np hard problem to guide the low rank metric learning experimental results on several standard benchmarks demonstrate the effectiveness of our proposed rtml by comparing it with the state of the art transfer learning and metric learning algorithms

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

dense error correction via ell 1 minimization this paper studies the problem of recovering a sparse signal x n from highly corrupted linear measurements y ax e m where e is an unknown error vector whose nonzero entries may be unbounded motivated by an observation from face recognition in computer vision this paper proves that for highly correlated and possibly overcomplete dictionaries a any sufficiently sparse signal x can be recovered by solving an l1 minimization problem min x 1 e 1 subject to y ax e more precisely if the fraction of the support of the error e is bounded away from one and the support of a is a very small fraction of the dimension m then as m becomes large the above l1 minimization succeeds for all signals x and almost all sign and support patterns of e this result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 of the observations corrupted the proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of independent identically distributed i i d gaussian vectors with nonzero mean and small variance dubbed the cross and bouquet cab model simulations and experiments corroborate the findings and suggest extensions to the result

saliency detection by multitask sparsity pursuit this paper addresses the problem of detecting salient areas within natural images we shall mainly study the problem under unsupervised setting i e saliency detection without learning from labeled images a solution of multitask sparsity pursuit is proposed to integrate multiple types of features for detecting saliency collaboratively given an image described by multiple features its saliency map is inferred by seeking the consistently sparse elements from the joint decompositions of multiple feature matrices into pairs of low rank and sparse matrices the inference process is formulated as a constrained nuclear norm and as an l2 1 norm minimization problem which is convex and can be solved efficiently with an augmented lagrange multiplier method compared with previous methods which usually make use of multiple features by combining the saliency maps obtained from individual features the proposed method seamlessly integrates multiple features to produce jointly the saliency map with a single inference step and thus produces more accurate and reliable results in addition to the unsupervised setting the proposed method can be also generalized to incorporate the top down priors obtained from supervised environment extensive experiments well validate its superiority over other state of the art methods

sparse canonical temporal alignment with deep tensor decomposition for action recognition in this paper we solve three problems in action recognition sub action multi subject and multi modality by reducing the diversity of intra class samples the main stage contains canonical temporal alignment and key frames selection as we know temporal alignment aims to reduce the diversity of intra class samples however dense frames may yield misalignment or overlapped alignment and decrease recognition performance to overcome this problem we propose a sparse canonical temporal alignment scta method which selects and aligns key frames from two sequences to reduce diversity to extract better features from the key frames we propose a deep non negative tensor factorization dntf method to find a tensor subspace integrated with scta scheme first we model an action sequence as a third order tensor with spatiotemporal structure then we design a dntf scheme to find a tensor subspace in both spatial and temporal directions particularly in the first layer the original tensor is decomposed into two low rank tensors by ntf and in the second layer each low rank tensor is further decomposed by tensor train for time efficiency finally our framework composed of scta and dntf could solve the three problems and extract effective features for action recognition experiments on synthetic data msrdailyactivity3d and msractionpairs data sets show that our method works better than competitive methods in terms of accuracy

algebraic clustering of affine subspaces subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition prior work has studied this problem using algebraic iterative statistical low rank and sparse representation techniques while these methods have been applied to both linear and affine subspaces theoretical results have only been established in the case of linear subspaces for example algebraic subspace clustering asc is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal in this paper we study in a rigorous fashion the properties of asc in the case of affine subspaces using notions from algebraic geometry we prove that the homogenization trick which embeds points in a union of affine subspaces into points in a union of linear subspaces preserves the general position of the points and the transversality of the union of subspaces in the embedded space thus establishing the correctness of asc for affine subspaces

tree structured nuclear norm approximation with applications to robust face recognition structured sparsity as an extension of standard sparsity has shown the outstanding performance when dealing with some highly correlated variables in computer vision and pattern recognition however the traditional mixed l1 l2 or l1 text l infty group norm becomes weak in characterizing the internal structure of each group since they cannot alleviate the correlations between variables recently nuclear norm has been validated to be useful for depicting a spatially structured matrix variable it considers the global structure of the matrix variable but overlooks the local structure to combine the advantages of structured sparsity and nuclear norm this paper presents a tree structured nuclear norm approximation tsna model assuming that the representation residual with tree structured prior is a random matrix variable and follows a dependent matrix distribution the extended alternating direction method of multipliers is utilized to solve the proposed model an efficient bound condition based on the extended restricted isometry constants is provided to show the exact recovery of the proposed model under the given noisy case in addition tsna is connected with some newest methods such as sparse representation based classifier nuclear l1 norm joint regression and nuclear norm based matrix regression which can be regarded as the special cases of tsna experiments with face reconstruction and recognition demonstrate the benefits of tsna over other approaches

robust bilinear factorization with missing and grossly corrupted observations recovering low rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics machine learning computer vision as well as signal and image processing in theory this problem can be solved by the natural convex joint mixed relaxations i e l 1 norm and trace norm under certain conditions however all current provable algorithms suffer from superlinear per iteration cost which severely limits their applicability to large scale problems in this paper we propose a scalable provable and structured robust bilinear factorization rbf method to recover low rank and sparse matrices from missing and grossly corrupted data i e robust matrix completion rmc or incomplete and grossly corrupted measurements i e compressive principal component pursuit cpcp specifically we first present two small scale matrix trace norm regularized bilinear factorization models for rmc and cpcp problems in which repetitively calculating svd of a large scale matrix is replaced by updating two much smaller factor matrices then we apply the alternating direction method of multipliers admm to efficiently solve the rmc problems finally we provide the convergence analysis of our algorithm and extend it to address general cpcp problems experimental results verified both the efficiency and effectiveness of our method compared with the state of the art methods

learning fast low rank projection for image classification rooted in a basic hypothesis that a data matrix is strictly drawn from some independent subspaces the low rank representation lrr model and its variations have been successfully applied in various image classification tasks however this hypothesis is very strict to the lrr model as it cannot always be guaranteed in real images moreover the hypothesis also prevents the sub dictionaries of different subspaces from collaboratively representing an image fortunately in supervised image classification low rank signal can be extracted from the independent label subspaces ils instead of the independent image subspaces iis therefore this paper proposes a projective low rank representation plr model by directly training a projective function to approximate the lrr derived from the labels to the best of our knowledge plr is the first attempt to use the ils hypothesis to relax the rigorous iis hypothesis in the lrr models we further prove a low rank effect that the representations learned by plr have high intraclass similarities and large interclass differences which are beneficial to the classification tasks the effectiveness of our proposed approach is validated by the experimental results on three databases

restricted isometry property of gaussian random projection for finite set of subspaces dimension reduction plays an essential role when decreasing the complexity of solving large scale problems the well known johnson lindenstrauss jl lemma and restricted isometry property rip admit the use of random projection to reduce the dimension while keeping the euclidean distance which leads to the boom of compressed sensing and the field of sparsity related signal processing recently successful applications of sparse models in computer vision and machine learning have increasingly hinted that the underlying structure of high dimensional data looks more like a union of subspaces in this paper motivated by jl lemma and an emerging field of compressed subspace clustering we study for the first time the rip of gaussian random matrices for the compression of two subspaces based on the generalized projection f norm distance we theoretically prove that with high probability the affinity or distance between two projected subspaces are concentrated around their estimates when the ambient dimension after projection is sufficiently large the affinity and distance between two subspaces almost remain unchanged after random projection numerical experiments verify the theoretical work

domain class consistency based transfer learning for image classification across domains distribution mismatch between the modeling data and the query data is a known domain adaptation issue in machine learning to this end in this paper we propose a l2 1 norm based discriminative robust kernel transfer learning dktl method for high level recognition tasks the key idea is to realize robust domain transfer by simultaneously integrating domain class consistency dcc metric based discriminative subspace learning kernel learning in reproduced kernel hilbert space and representation learning between source and target domain the dcc metric includes two properties domain consistency used to measure the between domain distribution discrepancy and class consistency used to measure the within domain class separability the essential objective of the proposed transfer learning method is to maximize the dcc metric which is equivalently to minimize the domain class inconsistency dcic such that domain distribution mismatch and class inseparability are well formulated and unified simultaneously the merits of the proposed method include 1 the robust sparse coding selects a few valuable source data with noises outliers removed during knowledge transfer and 2 the proposed dcc metric can pursue more discriminative subspaces of different domains as a result the maximum class separability is also well guaranteed extensive experiments on a number of visual datasets demonstrate the superiority of the proposed method over other state of the art domain adaptation and transfer learning methods

structured sparse representation with union of data driven linear and multilinear subspaces model for compressive video sampling the standard compressive sensing cs theory can be improved for robust recovery with fewer measurements under the assumption that signals lie on a union of subspaces uos however the uos model is restricted to specific types of signal regularities with predetermined topology for subspaces this paper proposes a generalized model which adaptively decomposes signals into a union of data driven subspaces uods for structured sparse representation the proposed uods model leverages subspace clustering to derive the optimal structures and bases for the subspaces conditioned on the sample signals for multidimensional signals with various statistics it supports linear and multilinear subspace learning for compressive sampling as an improvement for generic cs model the basis which represents the sparsity of sample signals is adaptively generated via linear subspace learning method furthermore a generalized model with multilinear subspace learning is considered for cs to avoid vectorization of high dimensional signals in comparison to uos the uods model requires fewer degrees of freedom for a desirable recovery quality experimental results demonstrate that the proposed model for video sampling is promising and applicable

robust elastic net subspace representation recently finding the low dimensional structure of high dimensional data has gained much attention given a set of data points sampled from a single subspace or a union of subspaces the goal is to learn or capture the underlying subspace structure of the data set in this paper we propose elastic net subspace representation a new subspace representation framework using elastic net regularization of singular values due to the strong convexity enforced by elastic net the proposed method is more stable and robust in the presence of heavy corruptions compared with existing lasso type rank minimization approaches for discovering a single low dimensional subspace we propose a computationally efficient low rank factorization algorithm called facten using a property of the nuclear norm and the augmented lagrangian method then clusten is proposed to handle the general case in which the data samples are drawn from a union of multiple subspaces for joint subspace clustering and estimation the proposed algorithms are applied to a number of subspace representation problems to evaluate the robustness and efficiency under various noisy conditions and experimental results show the benefits of the proposed method compared with existing methods