Most high-dimensional matrix recovery problems are studied under the assumption that the target matrix has certain intrinsic structures. For image data related matrix recovery problems, approximate low-rankness and smoothness are the two most commonly imposed structures. For approximately low-rank matrix recovery, the robust principal component analysis (PCA) is well-studied and proved to be effective. For smooth matrix problem, 2d fused Lasso and other total variation based approaches have played a fundamental role. Although both low-rankness and smoothness are key assumptions for image data analysis, the two lines of research, however, have very limited interaction. Motivated by taking advantage of both features, we in this paper develop a framework named projected robust PCA (PRPCA), under which the low-rank matrices are projected onto a space of smooth matrices. Consequently, a large class of image matrices can be decomposed as a low-rank and smooth component plus a sparse component. A key advantage of this decomposition is that the dimension of the core low-rank component can be significantly reduced. Consequently, our framework is able to address a problematic bottleneck of many low-rank matrix problems: singular value decomposition (SVD) on large matrices. Theoretically, we provide explicit statistical recovery guarantees of PRPCA and include classical robust PCA as a special case.
We study a practical algorithm for sparse principal component analysis (PCA) of incomplete and noisy data. Our algorithm is based on the semidefinite program (SDP) relaxation of the non-convex $l_1$-regularized PCA problem. We provide theoretical and experimental evidence that SDP enables us to exactly recover the true support of the sparse leading eigenvector of the unknown true matrix, despite only observing an incomplete (missing uniformly at random) and noisy version of it. We derive sufficient conditions for exact recovery, which involve matrix incoherence, the spectral gap between the la…
We analyze a practical algorithm for sparse PCA on incomplete and noisy data under a general non-random sampling scheme. The algorithm is based on a semidefinite relaxation of the $\ell_1$-regularized PCA problem. We provide theoretical justification that under certain conditions, we can recover the support of the sparse leading eigenvector with high probability by obtaining a unique solution. The conditions involve the spectral gap between the largest and second-largest eigenvalues of the true data matrix, the magnitude of the noise, and the structural properties of the observed entries. The …
Sparse Principal Component Analysis (PCA) is a prevalent tool across a plethora of subfields of applied statistics. While several results have characterized the recovery error of the principal eigenvectors, these are typically in spectral or Frobenius norms. In this paper, we provide entrywise $\ell_{2,\infty}$ bounds for Sparse PCA under a general high-dimensional subgaussian design. In particular, our results hold for any algorithm that selects the correct support with high probability, those that are sparsistent. Our bound improves upon known results by providing a finer characterization of…
We address the problem of defining a group sparse formulation for Principal Components Analysis (PCA) - or its equivalent formulations as Low Rank approximation or Dictionary Learning problems - which achieves a compromise between maximizing the variance explained by the components and promoting sparsity of the loadings. So we propose first a new definition of the variance explained by non necessarily orthogonal components, which is optimal in some aspect and compatible with the principal components situation. Then we use a specific regularization of this variance by the group-$\ell_{1}$ norm …
Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first $K$ principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank $K$ projections. Thus, PCA provides optimal low-rank representations of data for least-squares linear regression under standard modeling assumptions. On the other hand, when the loss function for a prediction problem is not the l…
In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. Specifically, the method achieves polynomial time complexity for certain convex cones equipped with fast projection such as the monotone cone. It attains a small error when the noisy matrix has a small cone-restricted operator norm. We supplement the above results with a minimax lower bound of the error under t…
University of Sheffield（谢菲尔德大学）；Sheffield Institute for Translational Neuroscience（谢菲尔德转化神经科学研究所）
Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other actors, for example sparse conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, wh…
Recovering intrinsic low dimensional subspaces from data distributed on them is a key preprocessing step to many applications. In recent years, there has been a lot of work that models 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-for…
Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel…
One technique to visualize the training of neural networks is to perform PCA on the parameters over the course of training and to project to the subspace spanned by the first few PCA components. In this paper we compare this technique to the PCA of a high dimensional random walk. We compute the eigenvalues and eigenvectors of the covariance of the trajectory and prove that in the long trajectory and high dimensional limit most of the variance is in the first few PCA components, and that the projection of the trajectory onto any subspace spanned by PCA components is a Lissajous curve. We genera…
We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, e…
Iowa State University（爱荷华州立大学）；Université de La Rochelle（拉罗谢尔大学）；University of Warwick（华威大学）
PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional…
We introduce a novel framework for an approxi- mate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherenc…
Principal component analysis (PCA) is a popular method for projecting data onto uncorrelated components in lower dimension, although the optimal number of components is not specified. Likewise, multiple signal classification (MUSIC) algorithm is a popular PCA-based method for estimating directions of arrival (DOAs) of sinusoidal sources, yet it requires the number of sources to be known a priori. The accurate estimation of the number of sources is hence a crucial issue for performance of these algorithms. In this paper, we will show that both PCA and MUSIC actually return the exact joint maxim…
Huazhong University of Science and Technology（华中科技大学）；University of Technology Sydney（悉尼科技大学）
Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label inf…
Random projection is widely used as a method of dimension reduction. In recent years, its combination with standard techniques of regression and classification has been explored. Here we examine its use with principal component analysis (PCA) and subspace detection methods. Specifically, we show that, under appropriate conditions, with high probability the magnitude of the residuals of a PCA analysis of randomly projected data behaves comparably to that of the residuals of a similar PCA analysis of the original data. Our results indicate the feasibility of applying subspace-based anomaly detec…
Principal Component Analysis (PCA) is a method for estimating a subspace given noisy samples. It is useful in a variety of problems ranging from dimensionality reduction to anomaly detection and the visualization of high dimensional data. PCA performs well in the presence of moderate noise and even with missing data, but is also sensitive to outliers. PCA is also known to have a phase transition when noise is independent and identically distributed; recovery of the subspace sharply declines at a threshold noise variance. Effective use of PCA requires a rigorous understanding of these behaviors…
University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Google Brain（谷歌大脑）
We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly Q…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex factorized optimization problem. The goal of this problem is to provide a general theoretical framework for understanding when such methods work well, and to characterize the nature of the resulting fixed point. We provide a simple set of conditions under which projected gradient descent…
We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose…
We propose Selective Multiple Power Iterations (SMPI), a new algorithm to address the important Tensor PCA problem that consists in recovering a spike $\bf{v_0}^{\otimes k}$ corrupted by a Gaussian noise tensor $\bf{Z} \in (\mathbb{R}^n)^{\otimes k}$ such that $\bf{T}=\sqrt{n} β\bf{v_0}^{\otimes k} + \bf{Z}$ where $β$ is the signal-to-noise ratio (SNR). SMPI consists in generating a polynomial number of random initializations, performing a polynomial number of symmetrized tensor power iterations on each initialization, then selecting the one that maximizes $\langle \bf{T}, \bf{v}^{\otimes k} \…
University of California, Irvine（加利福尼亚大学欧文分校）；Duke University（杜克大学）；Google Research（谷歌研究院）
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA…
Principal component analysis (PCA) is a well-established method commonly used to explore and visualise data. A classical PCA model is the fixed effect model where data are generated as a fixed structure of low rank corrupted by noise. Under this model, PCA does not provide the best recovery of the underlying signal in terms of mean squared error. Following the same principle as in ridge regression, we propose a regularised version of PCA that boils down to threshold the singular values. Each singular value is multiplied by a term which can be seen as the ratio of the signal variance over the t…
University of Tsukuba（筑波大学）；The Institute of Statistical Mathematics（统计数理研究所）
Principal component analysis (PCA) is a widely used method for data processing, such as for dimension reduction and visualization. Standard PCA is known to be sensitive to outliers, and thus, various robust PCA methods have been proposed. It has been shown that the robustness of many statistical methods can be improved using mode estimation instead of mean estimation, because mode estimation is not significantly affected by the presence of outliers. Thus, this study proposes a modal principal component analysis (MPCA), which is a robust PCA method based on mode estimation. The proposed method …
Carnegie Mellon University（卡内基梅隆大学）；National University of Singapore（新加坡国立大学）；Peking University（北京大学）；Shanghai Jiao Tong University（上海交通大学）；AI Institute（360人工智能研究院）
The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2018a] is an elegant extension of Robust PCA with a similar tight recovery bound, it is natural to solve other low rank tensor recovery problems extended from the matrix cases. However, the extensions and proofs are generally tedious. The general atomic norm provides a unified vie…
Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional data drawn from a low-dimensional subspace and degraded by heteroscedastic noise. We provide simplified expressions for the asymptotic PCA recovery of the underlying subspace, subspace a…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing glo…
We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existing algorithms cannot account for simultaneous element-wise and column-wise corruptions. In this paper, a new robust PCA algorithm that is robust to simultaneous types of corruption is proposed. Our approach hinges on the sparse approximation of a sparsely corrupted column so that…
In datasets where the number of parameters is fixed and the number of samples is large, principal component analysis (PCA) is a powerful dimension reduction tool. However, in many contemporary datasets, when the number of parameters is comparable to the sample size, PCA can be misleading. A closely related problem is the following: is it possible to recover a rank-one matrix in the presence of a large amount of noise? In both situations, there is a phase transition in the eigen-structure of the matrix.
This work obtains novel finite sample guarantees for Principal Component Analysis (PCA). These hold even when the corrupting noise is non-isotropic, and a part (or all of it) is data-dependent. Because of the latter, in general, the noise and the true data are correlated. The results in this work are a significant improvement over those given in our earlier work where this "correlated-PCA" problem was first studied. In fact, in certain regimes, our results imply that the sample complexity required to achieve subspace recovery error that is a constant fraction of the noise level is near-optimal…
This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employs principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accu…
Commonly used in computer vision and other applications, robust PCA represents an algorithmic attempt to reduce the sensitivity of classical PCA to outliers. The basic idea is to learn a decomposition of some data matrix of interest into low rank and sparse components, the latter representing unwanted outliers. Although the resulting optimization problem is typically NP-hard, convex relaxations provide a computationally-expedient alternative with theoretical support. However, in practical regimes performance guarantees break down and a variety of non-convex alternatives, including Bayesian-ins…
In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based spa…
Sparse PCA: Phase Transitions in the Critical Regime
Stern School of Business, New York University（纽约大学斯特恩商学院）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）
We consider the problem of high-dimensional misspecified phase retrieval. This is where we have an $s$-sparse signal vector $\mathbf{x}_*$ in $\mathbb{R}^n$, which we wish to recover using sampling vectors $\textbf{a}_1,\ldots,\textbf{a}_m$, and measurements $y_1,\ldots,y_m$, which are related by the equation $f(\left<\textbf{a}_i,\textbf{x}_*\right>) = y_i$. Here, $f$ is an unknown link function satisfying a positive correlation with the quadratic function. This problem was analyzed in a recent paper by Neykov, Wang and Liu, who provided recovery guarantees for a two-stage algorithm with samp…
We propose a generic framework based on a new stochastic variance-reduced gradient descent algorithm for accelerating nonconvex low-rank matrix recovery. Starting from an appropriate initial estimator, our proposed algorithm performs projected gradient descent based on a novel semi-stochastic gradient specifically designed for low-rank matrix recovery. Based upon the mild restricted strong convexity and smoothness conditions, we derive a projected notion of the restricted Lipschitz continuous gradient property, and prove that our algorithm enjoys linear convergence rate to the unknown low-rank…
We consider the problem of outlier robust PCA (OR-PCA) where the goal is to recover principal directions despite the presence of outlier data points. That is, given a data matrix $M^*$, where $(1-α)$ fraction of the points are noisy samples from a low-dimensional subspace while $α$ fraction of the points can be arbitrary outliers, the goal is to recover the subspace accurately. Existing results for \OR-PCA have serious drawbacks: while some results are quite weak in the presence of noise, other results have runtime quadratic in dimension, rendering them impractical for large scale applications…
Low-rank matrix recovery problems involving high-dimensional and heterogeneous data appear in applications throughout statistics and machine learning. The contribution of this paper is to establish the fundamental limits of recovery for a broad class of these problems. In particular, we study the problem of estimating a rank-one matrix from Gaussian observations where different blocks of the matrix are observed under different noise levels. In the setting where the number of blocks is fixed while the number of variables tends to infinity, we prove asymptotically exact formulas for the minimum …
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）
Tensor robust principal component analysis (TRPCA) is a fundamental model in machine learning and computer vision. Recently, tensor train (TT) decomposition has been verified effective to capture the global low-rank correlation for tensor recovery tasks. However, due to the large-scale tensor data in real-world applications, previous TRPCA models often suffer from high computational complexity. In this letter, we propose an efficient TRPCA under hybrid model of Tucker and TT. Specifically, in theory we reveal that TT nuclear norm (TTNN) of the original big tensor can be equivalently converted …
Wisconsin Institute for Discovery（威斯康星发现研究所）；University of Wisconsin - Madison（威斯康星大学麦迪逊分校）；University of Washington（华盛顿大学）
Empirical evidence suggests that for a variety of overparameterized nonlinear models, most notably in neural network training, the growth of the loss around a minimizer strongly impacts its performance. Flat minima -- those around which the loss grows slowly -- appear to generalize well. This work takes a step towards understanding this phenomenon by focusing on the simplest class of overparameterized nonlinear models: those arising in low-rank matrix recovery. We analyze overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and single hidden layer neural net…
Microsoft Corporation（微软公司）；Conduent Labs India（Conduent印度实验室）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of California Irvine（加利福尼亚大学尔湾分校）；Xerox Research Centre India（施乐印度研究中心）
The robust PCA problem, wherein, given an input data matrix that is the superposition of a low-rank matrix and a sparse matrix, we aim to separate out the low-rank and sparse components, is a well-studied problem in machine learning. One natural question that arises is that, as in the inductive setting, if features are provided as input as well, can we hope to do better? Answering this in the affirmative, the main goal of this paper is to study the robust PCA problem while incorporating feature information. In contrast to previous works in which recovery guarantees are based on the convex rela…
We develop an efficient algorithm for weak recovery in a robust version of the stochastic block model. The algorithm matches the statistical guarantees of the best known algorithms for the vanilla version of the stochastic block model. In this sense, our results show that there is no price of robustness in the stochastic block model. Our work is heavily inspired by recent work of Banks, Mohanty, and Raghavendra (SODA 2021) that provided an efficient algorithm for the corresponding distinguishing problem. Our algorithm and its analysis significantly depart from previous ones for robust recovery…
Mathematical Institute for Data Science（数据科学数学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）
Robust subspace recovery (RSR) is a fundamental problem in robust representation learning. Here we focus on a recently proposed RSR method termed Dual Principal Component Pursuit (DPCP) approach, which aims to recover a basis of the orthogonal complement of the subspace and is amenable to handling subspaces of high relative dimension. Prior work has shown that DPCP can provably recover the correct subspace in the presence of outliers, as long as the true dimension of the subspace is known. We show that DPCP can provably solve RSR problems in the {\it unknown} subspace dimension regime, as long…
Microsoft Research（微软研究院）；The University of California at Irvine（加州大学欧文分校）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
We propose a new method for robust PCA -- the task of recovering a low-rank matrix from sparse corruptions that are of unknown value and support. Our method involves alternating between projecting appropriate residuals onto the set of low-rank matrices, and the set of sparse matrices; each projection is {\em non-convex} but easy to compute. In spite of this non-convexity, we establish exact recovery of the low-rank matrix, under the same conditions that are required by existing methods (which are based on convex optimization). For an $m \times n$ input matrix ($m \leq n)$, our method has a run…
We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + λxx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + βxx^\top)$, respectively). Prior work has shown that when the signal-to-noise ratio ($λ$ or $β\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = ρ$, it is possible to recover $x$ in polynomial time if $ρ\less…
Various problems in data analysis and statistical genetics call for recovery of a column-sparse, low-rank matrix from noisy observations. We propose ReFACTor, a simple variation of the classical Truncated Singular Value Decomposition (TSVD) algorithm. In contrast to previous sparse principal component analysis (PCA) algorithms, our algorithm can provably reveal a low-rank signal matrix better, and often significantly better, than the widely used TSVD, making it the algorithm of choice whenever column-sparsity is suspected. Empirically, we observe that ReFACTor consistently outperforms TSVD eve…
Regularized variants of Principal Components Analysis, especially Sparse PCA and Functional PCA, are among the most useful tools for the analysis of complex high-dimensional data. Many examples of massive data, have both sparse and functional (smooth) aspects and may benefit from a regularization scheme that can capture both forms of structure. For example, in neuro-imaging data, the brain's response to a stimulus may be restricted to a discrete region of activation (spatial sparsity), while exhibiting a smooth response within that region. We propose a unified approach to regularized PCA which…
Department of Electrical and Computer Engineering, University of Minnesota – Twin Cities（明尼苏达大学双城分校电气与计算机工程系）
Localizing targets of interest in a given hyperspectral (HS) image has applications ranging from remote sensing to surveillance. This task of target detection leverages the fact that each material/object possesses its own characteristic spectral response, depending upon its composition. As $\textit{signatures}$ of different materials are often correlated, matched filtering based approaches may not be appropriate in this case. In this work, we present a technique to localize targets of interest based on their spectral signatures. We also present the corresponding recovery guarantees, leveraging…
Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace chan…
University of Minnesota（明尼苏达大学）；Princeton University（普林斯顿大学）
We consider the task of localizing targets of interest in a hyperspectral (HS) image based on their spectral signature(s), by posing the problem as two distinct convex demixing task(s). With applications ranging from remote sensing to surveillance, this task of target detection leverages the fact that each material/object possesses its own characteristic spectral response, depending upon its composition. However, since $\textit{signatures}$ of different materials are often correlated, matched filtering-based approaches may not be apply here. To this end, we model a HS image as a superposition …
