In this short note we extend some of the recent results on matrix completion under the assumption that the columns of the matrix can be grouped (clustered) into subspaces (not necessarily disjoint or independent). This model deviates from the typical assumption prevalent in the literature dealing with compression and recovery for big-data applications. The results have a direct bearing on the problem of subspace clustering under missing or incomplete information.
Boston University（波士顿大学）；University of Wisconsin - Madison（威斯康星大学麦迪逊分校）
This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n x N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank <= r < n, and assume N >> kn.…
In applications ranging from communications to genetics, signals can be modeled as lying in a union of subspaces. Under this model, signal coefficients that lie in certain subspaces are active or inactive together. The potential subspaces are known in advance, but the particular set of subspaces that are active (i.e., in the signal support) must be learned from measurements. We show that exploiting knowledge of subspaces can further reduce the number of measurements required for exact signal recovery, and derive universal bounds for the number of measurements needed. The bound is universal in …
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `complete' guarantee for ST-miss. This means that we can prove that under assumptions on only the algorithm inputs, the output subspace estimates are close to the true data subspaces at all times. Our guarantees hold under mild and easily interpretable assumptions, and allow the underl…
In this paper, we propose a simple algorithm to cluster nonnegative data lying in disjoint subspaces. We analyze its performance in relation to a certain measure of correlation between said subspaces. We use our clustering algorithm to develop a matrix completion algorithm which can outperform standard matrix completion algorithms on data matrices satisfying certain natural conditions.
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Wisconsin Institute for Discovery（威斯康星发现研究所）
This paper derives a perturbation bound on the optimal subspace estimator obtained from a subset of its canonical projections contaminated by noise. This fundamental result has important implications in matrix completion, subspace clustering, and related problems.
This work studies two interrelated problems - online robust PCA (RPCA) and online low-rank matrix completion (MC). In recent work by Candès et al., RPCA has been defined as a problem of separating a low-rank matrix (true data), $L:=[\ell_1, \ell_2, \dots \ell_{t}, \dots , \ell_{t_{\max}}]$ and a sparse matrix (outliers), $S:=[x_1, x_2, \dots x_{t}, \dots, x_{t_{\max}}]$ from their sum, $M:=L+S$. Our work uses this definition of RPCA. An important application where both these problems occur is in video analytics in trying to separate sparse foregrounds (e.g., moving objects) and slowly changing…
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In addition, varieties can be used to model a richer class of nonlinear quadratic and higher degree curves and surfaces. We…
In this paper, we consider clustering data that is assumed to come from one of finitely many pointed convex polyhedral cones. This model is referred to as the Union of Polyhedral Cones (UOPC) model. Similar to the Union of Subspaces (UOS) model where each data from each subspace is generated from a (unknown) basis, in the UOPC model each data from each cone is assumed to be generated from a finite number of (unknown) \emph{extreme rays}.To cluster data under this model, we consider several algorithms - (a) Sparse Subspace Clustering by Non-negative constraints Lasso (NCL), (b) Least squares ap…
Principal Components Analysis (PCA) is one of the most widely used dimension reduction techniques. Robust PCA (RPCA) refers to the problem of PCA when the data may be corrupted by outliers. Recent work by Cand{è}s, Wright, Li, and Ma defined RPCA as a problem of decomposing a given data matrix into the sum of a low-rank matrix (true data) and a sparse matrix (outliers). The column space of the low-rank matrix then gives the PCA solution. This simple definition has lead to a large amount of interesting new work on provably correct, fast, and practical solutions to RPCA. More recently, the dynam…
Many real world data sets exhibit an embedding of low-dimensional structure in a high-dimensional manifold. Examples include images, videos and internet traffic data. It is of great significance to reduce the storage requirements and computational complexity when the data dimension is high. Therefore we consider the problem of reconstructing a data stream from a small subset of its entries, where the data is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. We further consider tracking the change of the underlying subspace, which can be applied to applications s…
Marquette University（马凯特大学）；University of Wisconsin（威斯康星大学）；University of Michigan（密歇根大学）；University of Chicago（芝加哥大学）
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a secon…
Recent advances in matrix completion enable data imputation in full-rank matrices by exploiting low dimensional (nonlinear) latent structure. In this paper, we develop a new model for high rank matrix completion (HRMC), together with batch and online methods to fit the model and out-of-sample extension to complete new data. The method works by (implicitly) mapping the data into a high dimensional polynomial feature space using the kernel trick; importantly, the data occupies a low dimensional subspace in this feature space, even when the original data matrix is of full-rank. We introduce an ex…
In this note we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely many subspaces in infinite dimensional Hilbert spaces. This general approach allows us to unify many results derived recently in areas such as compressed sensing, affine rank minimisation and analog compressed sensing. Our main contribution is to show that a conceptually simple iterative projection algorithms is able to recover signals from a union of subspa…
Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on this theory, we demonstrate the efficiency of Subspace Clustering in fairly broad conditions. The insights and analysis techniques developed in this paper might also have implications for other random graph problems. Numerical experiments demonstrate the effectiveness of our theoretical study.
IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）；UBC（不列颠哥伦比亚大学）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；UC Berkeley（加州大学伯克利分校）
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation. In this paper, we consider matrix completion formulations designed to hit a target data-fitting error level provided by the user, and propose an algorithm called LR-BPDN that is able to exploit factorized formulations to solve the corresponding optimization problem. Since practitioners typically have strong prior knowledge about target error level, this inno…
We consider the phase retrieval problem for signals that belong to a union of subspaces. We assume that amplitude measurements of the signal of length $n$ are observed after passing it through a random $m \times n$ measurement matrix. We also assume that the signal belongs to the span of a single $d$-dimensional subspace out of $R$ subspaces, where $d\ll n$. We assume the knowledge of all possible subspaces, but the true subspace of the signal is unknown. We present an algorithm that jointly estimates the phase of the measurements and the subspace support of the signal. We discuss theoretical …
Wisconsin Institutes for Discovery（威斯康星发现研究院）；UW-Madison（威斯康星大学麦迪逊分校）；University of Michigan Ann Arbor（密歇根大学安娜堡分校）
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the challenge of matrix completion in the face of such nonlinearities. Given a few observations of a matrix that are obtained by applying a Lipschitz, monotonic function to a low rank matrix, our task is to estimate the remaining unobserved entries. We propose a novel matrix complet…
A well-known method for completing low-rank matrices based on convex optimization has been established by Cand{è}s and Recht. Although theoretically complete, the method may not entirely solve the low-rank matrix completion problem. This is because the method captures only the low-rankness property which gives merely a rough constraint that the data points locate on some low-dimensional subspace, but generally ignores the extra structures which specify in more detail how the data points locate on the subspace. Whenever the geometric distribution of the data points is not uniform, the coherence…
University of California, Davis（加州大学戴维斯分校）；University of Pennsylvania（宾夕法尼亚大学）
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually employed in order to improve the accuracy of matrix completion. In this paper, we propose a unified nonconvex optimization framework for matrix completion with linearly parameterized factors. In particular, by introducing a condition referred to as Correlated Parametric Factorization, we can conduct a unified geometric analysis for the nonconvex objective by…
Lower dimensional signal representation schemes frequently assume that the signal of interest lies in a single vector space. In the context of the recently developed theory of compressive sensing (CS), it is often assumed that the signal of interest is sparse in an orthonormal basis. However, in many practical applications, this requirement may be too restrictive. A generalization of the standard sparsity assumption is that the signal lies in a union of subspaces. Recovery of such signals from a small number of samples has been studied recently in several works. Here, we consider the problem o…
Matrix completion is a widely used technique for image inpainting and personalized recommender system, etc. In this work, we focus on accelerating the matrix completion using faster randomized singular value decomposition (rSVD). Firstly, two fast randomized algorithms (rSVD-PI and rSVD- BKI) are proposed for handling sparse matrix. They make use of an eigSVD procedure and several accelerating skills. Then, with the rSVD-BKI algorithm and a new subspace recycling technique, we accelerate the singular value thresholding (SVT) method in [1] to realize faster matrix completion. Experiments show t…
Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学机械化重点实验室）
Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random patterns. Specifically, a fundamental yet largely open question is to describe patterns that allow for unique or finitely many completions. This paper provides two such families of patterns for any rank. A key to achieving this is a novel formulation of low-rank matrix completion in terms of Plucker coordinates, the latter a traditional tool in computer vision. This…
Subspace clustering is a growing field of unsupervised learning that has gained much popularity in the computer vision community. Applications can be found in areas such as motion segmentation and face clustering. It assumes that data originate from a union of subspaces, and clusters the data depending on the corresponding subspace. In practice, it is reasonable to assume that a limited amount of labels can be obtained, potentially at a cost. Therefore, algorithms that can effectively and efficiently incorporate this information to improve the clustering model are desirable. In this paper, we …
We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and softw…
There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. This paper presents a clustering algorithm for high dimensional data that comes from a union of lower dimensional subspaces of equal and known dimensions. Such cases occur in many data clustering problems, such as …
Matrix completion refers to completing a low-rank matrix from a few observed elements of its entries and has been known as one of the significant and widely-used problems in recent years. The required number of observations for exact completion is directly proportional to rank and the coherency parameter of the matrix. In many applications, there might exist additional information about the low-rank matrix of interest. For example, in collaborative filtering, Netflix and dynamic channel estimation in communications, extra subspace information is available. More precisely in these applications,…
This paper studies the problem of completing a low-rank matrix from a few of its random entries with the aid of prior information. We suggest a strategy to incorporate prior information into the standard matrix completion procedure by maximizing the correlation between the original signal and the prior information. We also establish performance guarantees for the proposed method, which show that with suitable prior information, the proposed procedure can reduce the sample complexity of the standard matrix completion by a logarithmic factor. To illustrate the theory, we further analyze an impor…
We describe a new algorithm, termed subspace evolution and transfer (SET), for solving low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix, and outputs one low-rank matrix consistent with the given observations. The completion task is accomplished by searching for a column space on the Grassmann manifold that matches the incomplete observations. The SET algorithm consists of two parts -- subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of …
University of Central Florida（中佛罗里达大学）；Sharif University of Technology（谢里夫理工大学）
The problem of signal detection using a flexible and general model is considered. Due to applicability and flexibility of sparse signal representation and approximation, it has attracted a lot of attention in many signal processing areas. In this paper, we propose a new detection method based on sparse decomposition in a union of subspaces (UoS) model. Our proposed detector uses a dictionary that can be interpreted as a bank of matched subspaces. This improves the performance of signal detection, as it is a generalization for detectors. Low-rank assumption for the desired signals implies that …
An incoherent low-rank matrix can be efficiently reconstructed after observing a few of its entries at random, and then solving a convex program that minimizes the nuclear norm. In many applications, in addition to these entries, potentially valuable prior knowledge about the column and row spaces of the matrix is also available to the practitioner. In this paper, we incorporate this prior knowledge in matrix completion---by minimizing a weighted nuclear norm---and precisely quantify any improvements. In particular, we find in theory that reliable prior knowledge reduces the sample complexity …
In this paper we present a new model and an algorithm for unsupervised clustering of 2-D data such as images. We assume that the data comes from a union of multilinear subspaces (UOMS) model, which is a specific structured case of the much studied union of subspaces (UOS) model. For segmentation under this model, we develop Multilinear Subspace Clustering (MSC) algorithm and evaluate its performance on the YaleB and Olivietti image data sets. We show that MSC is highly competitive with existing algorithms employing the UOS model in terms of clustering performance while enjoying improvement in …
This paper studies a sparse signal recovery task in time-varying (time-adaptive) environments. The contribution of the paper to sparsity-aware online learning is threefold; first, a Generalized Thresholding (GT) operator, which relates to both convex and non-convex penalty functions, is introduced. This operator embodies, in a unified way, the majority of well-known thresholding rules which promote sparsity. Second, a non-convexly constrained, sparsity-promoting, online learning scheme, namely the Adaptive Projection-based Generalized Thresholding (APGT), is developed that incorporates the GT …
Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance for Gaussian windows. In this paper we develop a subspace completion algorithm, which enables stable reconstruction for a much wider choice of windows, including Gaussian windows. The combina…
This paper addresses the problem of simultaneous signal recovery and dictionary learning based on compressive measurements. Multiple signals are analyzed jointly, with multiple sensing matrices, under the assumption that the unknown signals come from a union of a small number of disjoint subspaces. This problem is important, for instance, in image inpainting applications, in which the multiple signals are constituted by (incomplete) image patches taken from the overall image. This work extends standard dictionary learning and block-sparse dictionary optimization, by considering compressive mea…
Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Various approaches have been proposed in the literature to identify time delays introduced by multipath environments. However, these methods either operate on the analog received signal, or require high sampling rates in order to achieve reasonable time resolution. In this paper, our goal is to develop a unified approach to time delay estimation from low rate samples of the output of a multipath channel. Our methods result in perfect recovery of the mu…
We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K-means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms a…
In this paper we present deterministic analysis of sufficient conditions for sparse subspace clustering under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. In this context we consider two cases, namely Case I when all the points are sampled at the same co-ordinates, and Case II when points are sampled at different locations. We show that results for Case I directly follow from several existing results in the literature, while results for Case II are not as straightforward and we provide a set of dual conditions under which, perfect clustering holds true. We …
A new algorithm, termed subspace evolution and transfer (SET), is proposed for solving the consistent matrix completion problem. In this setting, one is given a subset of the entries of a low-rank matrix, and asked to find one low-rank matrix consistent with the given observations. We show that this problem can be solved by searching for a column space that matches the observations. The corresponding algorithm consists of two parts -- subspace evolution and subspace transfer. In the evolution part, we use a line search procedure to refine the column space. However, line search is not guarantee…
In this paper we present deterministic conditions for success of sparse subspace clustering (SSC) under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. We consider two algorithms, which are variants of SSC with entry-wise zero-filling that differ in terms of the optimization problems used to find affinity matrix for spectral clustering. For both the algorithms, we provide deterministic conditions for any pattern of missing data such that perfect clustering can be achieved. We provide extensive sets of simulation results for clustering as well as completion of …
Traditional sampling theories consider the problem of reconstructing an unknown signal $x$ from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that $x$ lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which $x$ lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which $x$ lies in a sum of $k$ subspaces, chosen from a larger set of …
We study online robust matrix completion on graphs. At each iteration a vector with some entries missing is revealed and our goal is to reconstruct it by identifying the underlying low-dimensional subspace from which the vectors are drawn. We assume there is an underlying graph structure to the data, that is, the components of each vector correspond to nodes of a certain (known) graph, and their values are related accordingly. We give algorithms that exploit the graph to reconstruct the incomplete data, even in the presence of outlier noise. The theoretical properties of the algorithms are stu…
In this paper, we study the problem of estimating the direction of arrival (DOA) using a sparsely sampled uniform linear array (ULA). Based on an initial incomplete ULA measurement, our strategy is to choose a sparse subset of array elements for measuring the next snapshot. Then, we use a Hankel-structured matrix completion to interpolate for the missing ULA measurements. Finally, the source DOAs are estimated using a subspace method such as Prony on the fully recovered ULA. We theoretically provide a sufficient bound for the number of required samples (array elements) for perfect recovery. Th…
