The problem of consistently estimating the sparsity pattern of a vector $\betastar \in \real^\mdim$ based on observations contaminated by noise arises in various contexts, including subset selection in regression, structure estimation in graphical models, sparse approximation, and signal denoising. We analyze the behavior of $\ell_1$-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish a sharp relation between the problem dimension $\mdim$, the number $\spindex$ of non-zero elements in $\betastar$, and the number of observations $\numobs$ that are required for reliable recovery. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we establish existence and compute explicit values of thresholds $\ThreshLow$ and $\ThreshUp$ with the following properties: for any $ε> 0$, if $\numobs > 2 (\ThreshUp + ε) \log (\mdim - \spindex) + \spindex + 1$, then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for $\numobs < 2 (\ThreshLow - ε) \log (\mdim - \spindex) + \spindex + 1$, then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble, we show that $\ThreshLow = \ThreshUp = 1$, so that the threshold is sharp and exactly determined.
The problem of recovering the sparsity pattern of a fixed but unknown vector $β^* \in \real^p based on a set of $n$ noisy observations arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. Of interest are conditions on the model dimension $p$, the sparsity index $s$ (number of non-zero entries in $β^*$), and the number of observations $n$ that are necessary and/or sufficient to ensure asymptotically perfect recovery of the sparsity pattern. This paper focuses on the informatio…
We formulate sparse support recovery as a salient set identification problem and use information-theoretic analyses to characterize the recovery performance and sample complexity. We consider a very general model where we are not restricted to linear models or specific distributions. We state non-asymptotic bounds on recovery probability and a tight mutual information formula for sample complexity. We evaluate our bounds for applications such as sparse linear regression and explicitly characterize effects of correlation or noisy features on recovery performance. We show improvements upon previ…
We discuss a general notion of "sparsity structure" and associated recoveries of a sparse signal from its linear image of reduced dimension possibly corrupted with noise. Our approach allows for unified treatment of (a) the "usual sparsity" and "usual $\ell_1$ recovery," (b) block-sparsity with possibly overlapping blocks and associated block-$\ell_1$ recovery, and (c) low-rank-oriented recovery by nuclear norm minimization. The proposed recovery routines are natural extensions of the usual $\ell_1$ minimization used in Compressed Sensing. Specifically we present nullspace-type sufficient cond…
In this paper, we study a new notion of scaled minimaxity for sparse estimation in high-dimensional linear regression model. We present more optimistic lower bounds than the one given by the classical minimax theory and hence improve on existing results. We recover sharp results for the global minimaxity as a consequence of our study. Fixing the scale of the signal-to-noise ratio, we prove that the estimation error can be much smaller than the global minimax error. We construct a new optimal estimator for the scaled minimax sparse estimation. An optimal adaptive procedure is also described.
A trend in compressed sensing (CS) is to exploit structure for improved reconstruction performance. In the basic CS model, exploiting the clustering structure among nonzero elements in the solution vector has drawn much attention, and many algorithms have been proposed. However, few algorithms explicitly consider correlation within a cluster. Meanwhile, in the multiple measurement vector (MMV) model correlation among multiple solution vectors is largely ignored. Although several recently developed algorithms consider the exploitation of the correlation, these algorithms need to know a priori t…
We study the problem of consistently recovering the sparsity pattern of a regression parameter vector from correlated observations governed by deterministic missing data patterns using Lasso. We consider the case in which the observed dataset is censored by a deterministic, non-uniform filter. Recovering the sparsity pattern in datasets with deterministic missing structure can be arguably more challenging than recovering in a uniformly-at-random scenario. In this paper, we propose an efficient algorithm for missing value imputation by utilizing the topological property of the censorship filter…
This paper presents a novel narrowband interference (NBI) mitigation scheme for SC-FDMA systems. The proposed NBI cancellation scheme exploits the frequency domain sparsity of the unknown signal and adopts a low complexity Bayesian sparse recovery procedure. At the transmitter, a few randomly chosen sub-carriers are kept data free to sense the NBI signal at the receiver. Further, it is noted that in practice, the sparsity of the NBI signal is destroyed by a grid mismatch between NBI sources and the system under consideration. Towards this end, first an accurate grid mismatch model is presented…
Motivated by establishing theoretical foundations for various manifold learning algorithms, we study the problem of Mahalanobis distance (MD), and the associated precision matrix, estimation from high-dimensional noisy data. By relying on recent transformative results in covariance matrix estimation, we demonstrate the sensitivity of \MD~and the associated precision matrix to measurement noise, determining the exact asymptotic signal-to-noise ratio at which MD fails, and quantifying its performance otherwise. In addition, for an appropriate loss function, we propose an asymptotically optimal s…
We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the $l_2$ loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme.
We address the question of estimating Kullback-Leibler losses rather than squared losses in recovery problems where the noise is distributed within the exponential family. Inspired by Stein unbiased risk estimator (SURE), we exhibit conditions under which these losses can be unbiasedly estimated or estimated with a controlled bias. Simulations on parameter selection problems in applications to image denoising and variable selection with Gamma and Poisson noises illustrate the interest of Kullback-Leibler losses and the proposed estimators.
In the pivotal variable selection problem, we derive the exact non-asymptotic minimax selector over the class of all $s$-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2, and is also exact minimax with respect to the probability of wrong recovery. As a consequence, we establish explicit lower bounds under the monotone likelihood ratio property and we obtain a tigh…
Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $\ell_1$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements. Leveraging the low-rank property of the tensor, this paper propo…
This paper introduces an efficient sparse recovery approach for Polynomial Chaos (PC) expansions, which promotes the sparsity by breaking the dimensionality of the problem. The proposed algorithm incrementally explores sub-dimensional expansions for a sparser recovery, and shows success when removal of uninfluential parameters that results in a lower coherence for measurement matrix, allows for a higher order and/or sparser expansion to be recovered. The incremental algorithm effectively searches for the sparsest PC approximation, and not only can it decrease the prediction error, it can also …
In this paper, we study the problem of recovering a group sparse vector from a small number of linear measurements. In the past the common approach has been to use various "group sparsity-inducing" norms such as the Group LASSO norm for this purpose. By using the theory of convex relaxations, we show that it is also possible to use $\ell_1$-norm minimization for group sparse recovery. We introduce a new concept called group robust null space property (GRNSP), and show that, under suitable conditions, a group version of the restricted isometry property (GRIP) implies the GRNSP, and thus leads t…
This paper describes a simple framework for structured sparse recovery based on convex optimization. We show that many structured sparsity models can be naturally represented by linear matrix inequalities on the support of the unknown parameters, where the constraint matrix has a totally unimodular (TU) structure. For such structured models, tight convex relaxations can be obtained in polynomial time via linear programming. Our modeling framework unifies the prevalent structured sparsity norms in the literature, introduces new interesting ones, and renders their tightness and tractability argu…
We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions…
In this paper we consider asymptotically exact support recovery in the context of high dimensional and sparse Canonical Correlation Analysis (CCA). Our main results describe four regimes of interest based on information theoretic and computational considerations. In regimes of "low" sparsity we describe a simple, general, and computationally easy method for support recovery, whereas in a regime of "high" sparsity, it turns out that support recovery is information theoretically impossible. For the sake of information theoretic lower bounds, our results also demonstrate a non-trivial requirement…
Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the r…
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）；UPS IMT（图卢兹第三大学数学研究所）；Institut de Recherche Technologique Saint-Exupéry（圣埃克苏佩里技术研究所）
We consider a deep structured linear network under sparsity constraints. We study sharp conditions guaranteeing the stability of the optimal parameters defining the network. More precisely, we provide sharp conditions on the network architecture and the sample under which the error on the parameters defining the network scales linearly with the reconstruction error (i.e. the risk). Therefore, under these conditions, the weights obtained with a successful algorithms are well defined and only depend on the architecture of the network and the sample. The features in the latent spaces are stably d…
It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms L1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted L1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. …
An increasing number of applications is concerned with recovering a sparse matrix from noisy observations. In this paper, we consider the setting where each row of the unknown matrix is sparse. We establish minimax optimal rates of convergence for estimating matrices with row sparsity. A major focus in the present paper is on the derivation of lower bounds.
