robust subspace clustering with complex noise subspace clustering has important and wide applications in computer vision and pattern recognition it is a challenging task to learn low dimensional subspace structures due to complex noise existing in high dimensional data complex noise has much more complex statistical structures and is neither gaussian nor laplacian noise recent subspace clustering methods usually assume a sparse representation of the errors incurred by noise and correct these errors iteratively however large corruptions incurred by complex noise cannot be well addressed by these methods a novel optimization model for robust subspace clustering is proposed in this paper its objective function mainly includes two parts the first part aims to achieve a sparse representation of each high dimensional data point with other data points the second part aims to maximize the correntropy between a given data point and its low dimensional representation with other points correntropy is a robust measure so that the influence of large corruptions on subspace clustering can be greatly suppressed an extension of pairwise link constraints is also proposed as prior information to deal with complex noise half quadratic minimization is provided as an efficient solution to the proposed robust subspace clustering formulations experimental results on three commonly used data sets show that our method outperforms state of the art subspace clustering methods

solution to the inverse regulator problem for discrete time systems a complete solution is given for the inverse problem of lq optimal regulators for the discrete time multi input case the kalman equation for discrete time systems is derived to characterize the optimal regulators it is then shown that this equation holds for some quadratic weighting if and only if the feedback satisfies a condition corresponding to the circle criterion for continuous time systems the optimality criterion given here consists of this condition and an additional condition to ensure detectability an example is given to illustrate the results

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

hypermatrix factors for string and membrane junctions the adjoint representations of the lie algebras of the classical groups su n so n and sp n are respectively tensor antisymmetric and symmetric products of two vector spaces and hence are matrix representations we consider the analogous products of three vector spaces and study when they appear as summands in lie algebra decompositions the grading of the exceptional lie algebras provides such summands and provides representations of classical groups on hypermatrices the main natural application is a formal study of 3 junctions of strings and membranes generalizations are also considered

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

spectral clustering on multiple manifolds spectral clustering sc is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data though sc methods have been successfully applied to a large number of challenging clustering scenarios it is noteworthy that they will fail when there are significant intersections among different clusters in this paper based on the analysis that sc methods are able to work well when the affinity values of the points belonging to different clusters are relatively low we propose a new method called spectral multi manifold clustering smmc which is able to handle intersections in our model the data are assumed to lie on or close to multiple smooth low dimensional manifolds where some data manifolds are separated but some are intersecting then local geometric information of the sampled data is incorporated to construct a suitable affinity matrix finally spectral method is applied to this affinity matrix to group the data extensive experiments on synthetic as well as real datasets demonstrate the promising performance of smmc

innovation pursuit a new approach to subspace clustering in subspace clustering a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces this paper presents a new approach dubbed innovation pursuit ipursuit to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on their relative novelties we present two frameworks in which the idea of innovation pursuit is used to distinguish the subspaces underlying the first framework is an iterative method that finds the subspaces consecutively by solving a series of simple linear optimization problems each searching for a direction of innovation in the span of the data potentially orthogonal to all subspaces except for the one to be identified in one step of the algorithm a detailed mathematical analysis is provided establishing sufficient conditions for ipursuit to correctly cluster the data the proposed approach can provably yield exact clustering even when the subspaces have significant intersections it is shown that the complexity of the iterative approach scales only linearly in the number of data points and subspaces and quadratically in the dimension of the subspaces the second framework integrates ipursuit with spectral clustering to yield a new variant of spectral clustering based algorithms the numerical simulations with both real and synthetic data demonstrate that ipursuit can often outperform the state of the art subspace clustering algorithms more so for subspaces with significant intersections and that it significantly improves the state of the art result for subspace segmentation based face clustering

a revisit to inverse optimality of linear systems in this article we revisit the problem of inverse optimality for linear systems by applying certain explicit formulae for coprime matrix fraction descriptions cmfd of linear systems we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index compared to existing results in the literature the proposed condition is simpler to check and interpret moreover it reduces the redundancy in the solutions of the associated algebraic riccati equation are as a direct application of the proposed results we consider the problem of inverse optimality of observer based state feedback to be specific for the case where the state is not fully known we consider the inverse optimality problem of an observer based state feedback for the closed loop system augmented by an observer more precisely it is shown that the observer based state feedback is inverse optimal for the closed loop system with some general forms of cost functions on

generalized robustness of optimally of linear quadratic regulators robustness of the optimality of linear quadratic lq optimal regulators is studied it is shown that under a mild assumption optimality of lq regulators is preserved under small perturbations on both f and g matrices this is made possible by allowing modifications on both weighting parameters q and r and it also extends the existing results where perturbations on f only are allowed it is also observed that this modification leads to a possible change in gain margin a numerical example is discussed in detail to illustrate the results

cross standard form a solution to improve a given controller with h 2 or h specifications this paper introduces in cross standard form csf as a solution to the inverse optimal control problem that is the csf is a canonical standard problem whose unique h or h 2 optimal controller is a given controller from the control design point of view the general idea is to apply the csf to a given controller in order to set up a standard problem which can be completed to handle frequency domain h 2 or h specification the analytical formulation of the csf proposed in this paper can be applied to reduced full or augmented order compensators or two degree of freedom compensations numerical and academic examples are given

hankel hyperdeterminants rectangular jack polynomials and even powers of the vandermonde abstract we investigate the link between rectangular jack polynomials and hankel hyperdeterminants as an application we give an expression of the even power of the vandermonde in terms of jack polynomials

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

a deterministic analysis for lrr the recently proposed low rank representation lrr method has been empirically shown to be useful in various tasks such as motion segmentation image segmentation saliency detection and face recognition while potentially powerful lrr depends heavily on the configuration of its key parameter lambda in realistic environments where the prior knowledge about data is lacking however it is still unknown how to choose lambda in a suitable way even more there is a lack of rigorous analysis about the success conditions of the method and thus the significance of lrr is a little bit vague in this paper we therefore establish a theoretical analysis for lrr striving for figuring out under which conditions lrr can be successful and deriving a moderately good estimate to the key parameter lambda as well simulations on synthetic data points and experiments on real motion sequences verify our claims

a new approach to two view motion segmentation using global dimension minimization we present a new approach to rigid body motion segmentation from two views we use a previously developed nonlinear embedding of two view point correspondences into a 9 dimensional space and identify the different motions by segmenting lower dimensional subspaces in order to overcome nonuniform distributions along the subspaces whose dimensions are unknown we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation we propose a fast projected gradient algorithm for minimizing global dimension and thus segmenting motions from 2 views we develop an outlier detection framework around the proposed method and we present state of the art results on outlier free and outlier corrupted two view data for segmenting motion

new solution to the inverse regulator problem by the polynomial matrix method abstract this paper gives a new criterion for the inverse regulator problem in terms of polynomial matrix fractions the generalized kalman equation in polynomial matrix form leads to a necessary and sufficient condition for a stable feedback law to be optimal for some unknown quadratic performance index this criterion consists of i a symmetric factorizability of a polynomial matrix and ii a polynomial type criterion that guarantees detectability the relationship between the riccati and kalman equations is also established finally an example is given to illustrate the new result in contrast with the existing result

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

clustering with hypergraphs the case for large hyperedges the extension of conventional clustering to hypergraph clustering which involves higher order similarities instead of pairwise similarities is increasingly gaining attention in computer vision this is due to the fact that many clustering problems require an affinity measure that must involve a subset of data of size more than two in the context of hypergraph clustering the calculation of such higher order similarities on data subsets gives rise to hyperedges almost all previous work on hypergraph clustering in computer vision however has considered the smallest possible hyperedge size due to a lack of study into the potential benefits of large hyperedges and effective algorithms to generate them in this paper we show that large hyperedges are better from both a theoretical and an empirical standpoint we then propose a novel guided sampling strategy for large hyperedges based on the concept of random cluster models our method can generate large pure hyperedges that significantly improve grouping accuracy without exponential increases in sampling costs we demonstrate the efficacy of our technique on various higher order grouping problems in particular we show that our approach improves the accuracy and efficiency of motion segmentation from dense long term trajectories

face recognition via collaborative representation its discriminant nature and superposed representation collaborative representation methods such as sparse subspace clustering ssc and sparse representation based classification src have achieved great success in face clustering and classification by directly utilizing the training images as the dictionary bases in this paper we reveal that the superior performance of collaborative representation relies heavily on the sufficiently large class separability of the controlled face datasets such as extended yale b on the uncontrolled or undersampled dataset however collaborative representation suffers from the misleading coefficients of the incorrect classes to address this limitation inspired by the success of linear discriminant analysis lda we develop a superposed linear representation classifier slrc to cast the recognition problem by representing the test image in term of a superposition of the class centroids and the shared intra class differences in spite of its simplicity and approximation the slrc largely improves the generalization ability of collaborative representation and competes well with more sophisticated dictionary learning techniques on the experiments of ar and frgc databases enforced with the sparsity constraint slrc achieves the state of the art performance on feret database using single sample per person

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

robust subspace clustering via thresholding the problem of clustering noisy and incompletely observed high dimensional data points into a union of low dimensional subspaces and a set of outliers is considered the number of subspaces their dimensions and their orientations are assumed unknown we propose a simple low complexity subspace clustering algorithm which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points in other words the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance a statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect specifically our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level we furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be up to a log factor linear in the ambient dimension we also propose a simple scheme that provably detects outliers and we present numerical results on real and synthetic data

a geometric analysis of subspace clustering with outliers this paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes as is common in computer vision or unsupervised learning applications we do not know in advance how many subspaces there are nor do we have any information about their dimensions we develop a novel geometric analysis of an algorithm named sparse subspace clustering ssc 11 which signicantly broadens the range of problems where it is provably eective for instance we show that ssc can recover multiple subspaces each of dimension comparable to the ambient dimension we also prove that ssc can correctly cluster data points even when the subspaces of interest intersect further we develop an extension of ssc that succeeds when the data set is corrupted with possibly overwhelmingly many outliers underlying our analysis are clear geometric insights which may bear on other sparse recovery problems a numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods

quantum hyperdeterminants and hyper pfaffians the notion of generalized quantum monoids is introduced it is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hyper algebra in which the quantum determinant and quantum pfaffian are sent to the quantum hyperdeterminant and quantum hyper pfaffian respectively the quantum hyperdeterminant in even dimension is shown to be a q analog of cayley s first hyperdeterminant

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

a polynomial characterization of hypergraphs using the ihara zeta function the aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering to this end we develop a polynomial characterization for hypergraphs based on the ihara zeta function we investigate the flexibility of the polynomial coefficients for learning relational structures with different relational orders furthermore we develop an efficient method for computing the coefficient set our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities and thus can distinguish different relational orders which is prone to ambiguity if the hypergraph laplacian is used in our experimental evaluation we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph laplacian

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

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

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

a structured sparse plus structured low rank framework for subspace clustering and completion recent advances in a low rank matrix completion have enabled the exact recovery of incomplete data drawn from a low dimensional subspace of a high dimensional observation space however in many applications the data are drawn from multiple low dimensional subspaces without knowing which point belongs to which subspace in such cases using a single low dimensional subspace to complete the data may lead to erroneous results because the complete data matrix need not be low rank in this paper we propose a structured sparse plus structured low rank text s 3 lr optimization framework for clustering and completing data drawn from a union of low dimensional subspaces the proposed text s 3 lr framework exploits the fact that each point in a union of subspaces can be expressed as a sparse linear combination of all other points and that the matrix of the points within each subspace is low rank this framework leads to a nonconvex optimization problem which we solve efficiently by using a combination of a linearized alternating direction method of multipliers and spectral clustering in addition we discuss the conditions that guarantee the exact matrix completion in a union of subspaces experiments on synthetic data motion segmentation data and cancer gene data validate the effectiveness of the proposed approach

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

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

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

hyperdeterminantal expressions for jack functions of rectangular shapes abstract we derive a jacobi trudi type formula for jack functions of rectangular shapes in this formula we make use of a hyperdeterminant which is cayley s simple generalization of the determinant in addition after developing the general theory of hyperdeterminants we give summation formulas for schur functions involving hyperdeterminants and evaluate toeplitz type hyperdeterminants by using jack function theory

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

new algorithms for linear k matroid intersection and matroid k parity problems we present algorithms for thek matroid intersection problem and for the matroidk parity problem when the matroids are represented over the field of rational numbers andk 2 the computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids as an application we describe new polynomially solvable cases of thek dimensional assignment problem and of thek dimensional matching problem the algorithms use some new identities in multilinear algebra including the generalized binet cauchy formula and its analogue for the pfaffian these techniques extend known methods developed earlier fork 2

the fiedler vector of a laplacian tensor for hypergraph partitioning based on recent advances in spectral hypergraph theory l qi and z luo tensor anaysis spectral theory and special tensors siam philadelphia 2017 we explore the fiedler vector of an even uniform hypergraph which is the z eigenvector associated with the second smallest z eigenvalue of a normalized laplacian tensor arising from the hypergraph then we develop a novel tensor based spectral method for partitioning vertices of the hypergraph for this purpose we extend the normalized laplacian matrix of a simple graph to the normalized laplacian tensor of an even uniform hypergraph the corresponding fiedler vector is related to the cheeger constant of the hypergraph then we establish a feasible optimization algorithm to compute the fiedler vector according to the normalized laplacian tensor the convergence of the proposed algorithm and the probability of obtaining the fiedler vector of the hypergraph are analyzed theoretically finally preliminary numerical experiments illustrate that the new a

enhanced local subspace affinity for feature based motion segmentation we present a new motion segmentation algorithm the enhanced local subspace affinity elsa unlike local subspace affinity elsa is robust in a variety of conditions even without manual tuning of its parameters this result is achieved thanks to two improvements the first is a new model selection technique for the estimation of the trajectory matrix rank the second is an estimation of the number of motions based on the analysis of the eigenvalue spectrum of the symmetric normalized laplacian matrix results using the hopkins155 database and synthetic sequences are presented and compared with state of the art techniques

algebraic algorithms for matching and matroid problems we present new algebraic approaches for two well known combinatorial problems nonbipartite matching and matroid intersection our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms for nonbipartite matching we obtain a simple purely algebraic algorithm with running time o n omega where n is the number of vertices and omega is the matrix multiplication exponent this resolves the central open problem of mucha and sankowski 2004 for matroid intersection our algorithm has running time o nr omega 1 for matroids with n elements and rank r that satisfy some natural conditions

greedy feature selection for subspace clustering unions of subspaces provide a powerful generalization of single subspace models for collections of high dimensional data however learning multiple subspaces from data is challenging due to the fact that segmentation the identification of points that live in the same subspace and subspace estimation must be performed simultaneously recently sparse recovery methods were shown to provide a provable and robust strategy for exact feature selection efs recovering subsets of points from the ensemble that live in the same subspace in parallel with recent studies of efs with l1 minimization in this paper we develop sufficient conditions for efs with a greedy method for sparse signal recovery known as orthogonal matching pursuit omp following our analysis we provide an empirical study of feature selection strategies for signals living on unions of subspaces and characterize the gap between sparse recovery methods and nearest neighbor nn based approaches in particular we demonstrate that sparse recovery methods provide significant advantages over nn methods and that the gap between the two approaches is particularly pronounced when the sampling of subspaces in the data set is sparse our results suggest that omp may be employed to reliably recover exact feature sets in a number of regimes where nn approaches fail to reveal the subspace membership of points in the ensemble

hybrid linear modeling via local best fit flats we present a simple and fast geometric method for modeling data by a union of affine subspaces the method begins by forming a collection of local best fit affine subspaces i e subspaces approximating the data in local neighborhoods the correct sizes of the local neighborhoods are determined automatically by the jones 2 numbers we prove under certain geometric conditions that our method finds the optimal local neighborhoods the collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model we discuss applications to tracking based motion segmentation and clustering of faces under different illuminating conditions we give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the mnist handwritten digits data and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces

on successive pole assignment by linear quadratic optimal feedbacks abstract this paper studies a method of shifting poles of linear constant systems via lq optimal feedback by making use of a solution to the so called inverse regulator problem this method enables us to shift poles successively by pairs while maintaining the well known advantages of lq regulators polynomial fractional representation is effectively used both to characterize the maximal pole assignable area which clarifies the theoretical bound of the lq optimal pole shifting and to derive an algorithm a numerical example is given to illustrate the results

an efficient tree decomposition method for permanents and mixed discriminants abstract we present an efficient algorithm to compute permanents mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays tensors we describe the sparsity structure of an array in terms of a graph and we assume that its treewidth denoted as is small our algorithm requires o n 2 arithmetic operations to compute permanents and o n 2 n 3 for mixed discriminants and hyperdeterminants we finally show that mixed volume computation continues to be hard under bounded treewidth assumptions

a complete optimality condition in the inverse problem of optimal control a complete optimality condition in the standard inverse problem of optimal control for the multi input case is established the optimality of a feedback control law is characterized completely in terms of a new geometric condition as well as the well known return difference condition the new condition not only provides a better insight into the well known sensitivity reduction property of optimal control for the multi input case but also indicates an essential difference between the solutions of the single and multi input inverse problem

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

sparse subspace clustering algorithm theory and applications many real world problems deal with collections of high dimensional data such as images videos text and web documents dna microarray data and more often such high dimensional data lie close to low dimensional structures corresponding to several classes or categories to which the data belong in this paper we propose and study an algorithm called sparse subspace clustering to cluster data points that lie in a union of low dimensional subspaces the key idea is that among the infinitely many possible representations of a data point in terms of other points a sparse representation corresponds to selecting a few points from the same subspace this motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces since solving the sparse optimization program is in general np hard we consider a convex relaxation and show that under appropriate conditions on the arrangement of the subspaces and the distribution of the data the proposed minimization program succeeds in recovering the desired sparse representations the proposed algorithm is efficient and can handle data points near the intersections of subspaces another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances such as noise sparse outlying entries and missing entries by incorporating the model of the data into the sparse optimization program we demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real world problems of motion segmentation and face clustering

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

generalized projection based m estimator we propose a novel robust estimation algorithm the generalized projection based m estimator gpbm which does not require the user to specify any scale parameters the algorithm is general and can handle heteroscedastic data with multiple linear constraints for single and multicarrier problems the gpbm has three distinct stages scale estimation robust model estimation and inlier outlier dichotomy in contrast in its predecessor pbm each model hypotheses was associated with a different scale estimate for data containing multiple inlier structures with generally different noise covariances the estimator iteratively determines one structure at a time the model estimation can be further optimized by using grassmann manifold theory we present several homoscedastic and heteroscedastic synthetic and real world computer vision problems with single and multiple carriers

robust subspace clustering subspace clustering refers to the task of nding a multi subspace representation that best ts a collection of points taken from a high dimensional space this paper introduces an algorithm inspired by sparse subspace clustering ssc 25 to cluster noisy data and develops some novel theory demonstrating its correctness in particular the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation and on the number of samples per subspace synthetic as well as real data experiments complement our theoretical study illustrating our approach and demonstrating its eectiveness

noisy sparse subspace clustering this paper considers the problem of subspace clustering under noise specifically we study the behavior of sparse subspace clustering ssc when either adversarial or random noise is added to the unlabeled input data points which are assumed to be in a union of low dimensional subspaces we show that a modified version of ssc is provably effective in correctly identifying the underlying subspaces even with noisy data this extends theoretical guarantee of this algorithm to more practical settings and provides justification to the success of ssc in a class of real applications