Null Space Property Research Articles

A novel property to modify weighted [formula omitted] minimization for improved compressed sensing

Weighted l1 minimization schemes are common methods to achieve compressed sensing (CS). However, they fail in the presence of inaccurate prior knowledge or improper scaling of weights due to inappropriately assigned large weights causing large and destructive errors in signal recovery. This paper proposes a theory-based algorithm to identify and correct such destructive weights for each signal entry. The enhancement is achieved through a novel sparsity-inducing property (SIP) which establishes a necessary condition for successful signal recovery. SIP outperforms existing properties such as coherence, restricted isometry property, and nullspace property by indicating which signal entries fail to be recovered. This unique advantage enables us to correct destructive weights that do not satisfy the SIP condition, making signal recovery successful where it previously failed. Results from many numerical experiments demonstrate that our proposed method can improve the signal recovery capability, robustness, and stability of the weighted l1 minimization for a wide range of applications, including sparse and compressive signal recovery, noise-aware recovery, sparse error correction, fast image acquisition, and sub-Nyquist sampling.

The Perturbation Analysis of Nonconvex Low-Rank Matrix Robust Recovery.

In this article, we bring forward a completely perturbed nonconvex Schatten p -minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten- p null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten- p NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that p decreases 0 and for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition (Recht et al., 2010). In addition, we study the connection between RIP and Schatten- p NSP and discern that Schatten- p NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten p -minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

Sparse Signal Reconstruction: Sequential Convex Relaxation, Restricted Null Space Property, and Error Bounds

Sparse Signal Reconstruction: Sequential Convex Relaxation, Restricted Null Space Property, and Error Bounds

Analyses of the tail-[formula omitted] minimization for fast and enhanced sparse selections

We investigate the effectiveness and efficiency of the iterative tail-ℓ2 minimization (tail-ℓ2-min) technique for its sparse selection capabilities. We conduct profile analyses on the tail-ℓ2-min, establishing the equivalence of the tail-ℓ2-min problem to a two-stage profile ℓ2 formulation, both featuring analytical solutions. The tail null space property (NSP) of sensing matrix A is shown to be equivalent to the NSP of the newly defined profile matrix Ã. Besides the error bound analysis for the tail-ℓ2-min under the typical tail-NSP condition, a novel error bound of the tail-ℓ2-min formulation is also established without relying on NSP or restricted isometry property (RIP) assumptions. It merely contains tractable coefficients of A, and offers insights into successful recovery, with the observation of the convergent iterative procedure. Numerical studies and the applications to image reconstruction demonstrate the superiority and fast convergence of the tail-ℓ2 sparse solution over state-of-the-art sparse selection methodologies. The sparsity level of a signal that the tail-ℓ2 profile algorithm guarantees the recovery is around 41% higher than that of the basis pursuit algorithm. The analytical solutions of the tail-ℓ2 method at each iteration also ensure that the tail-ℓ2 sparse recovery process is notably fast, especially for high dimensions and high sparsity levels.

Robust sparse recovery with sparse Bernoulli matrices via expanders

Sparse binary matrices are of great interest in the field of sparse recovery, nonnegative compressed sensing, statistics in networks, and theoretical computer science. This class of matrices makes it possible to perform signal recovery with lower storage costs and faster decoding algorithms. In particular, Bernoulli (p) matrices formed by independent identically distributed (i.i.d.) Bernoulli (p) random variables are of practical relevance in the context of noise-blind recovery in nonnegative compressed sensing.In this work, we investigate the robust nullspace property of Bernoulli (p) matrices. Previous results in the literature establish that such matrices can accurately recover n-dimensional s-sparse vectors with m=O(sc(p)log⁡ens) measurements, where c(p)≤p is a constant dependent only on the parameter p. These results suggest that in the sparse regime, as p approaches zero, the (sparse) Bernoulli (p) matrix requires significantly more measurements than the minimal necessary, as achieved by standard isotropic subgaussian designs. However, we show that this is not the case.Our main result characterizes, for a wide range of sparsity levels s, the smallest p for which sparse recovery can be achieved with the minimal number of measurements. We also provide matching lower bounds to establish the optimality of our results and explore connections with the theory of invertibility of discrete random matrices and integer compressed sensing.

Frame-based block sparse compressed sensing via l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis

In this paper, we consider the frame-based block sparse signal recovery via a l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis method. A new kind of null space property based on the given dictionary D(block D-NSP) is proposed. It is proved that sensing matrices satisfying the block D-NSP is not just a sufficient and necessary condition for the l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis method to exactly recover signals that are block sparse in frame D, but also a sufficient and necessary condition for the l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis to stably recover signals which are block-compressible in frame D. To the best of our knowledge, this new property is the first sufficient and necessary condition for successful signal recovery via the l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis. In addition, we also characterize the theoretical performance of recovering signals via the l2/l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2/l_1$$\\end{document}-synthesis in the case of the measurements are disturbed.

Separation of Multicomponent Micro-Doppler Signal with Missing Samples

The problem of separating multicomponent micro-Doppler (m-D) signals is common in the field of radar signal processing. In some implementations, it is necessary to separate the multicomponent m-D signal that contains missing samples. To address this issue, an optimization model has been developed to recover and decompose multicomponent m-D signals with missing samples. To solve the underlying optimization problem, a two-algorithm-based alternate iteration framework is proposed. This method uses three techniques—the null space property, ridge regression method, and matching pursuit principle—to estimate the individual component, complex-valued differential operator, and regularization parameter. Finally, as shown by both simulation and measured data processing results, the proposed method can accurately separate the multicomponent m-D signal from incomplete data.

On the update of algebraic states during state estimation of differential–algebraic equation (DAE) systems

This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure.

A unified recovery of structured signals using atomic norm

Abstract In many applications, we seek to recover signals from linear measurements far fewer than the ambient dimension, given the signals have exploitable structures such as sparse vectors or low rank matrices. In this paper, we work in a general setting where signals are approximately sparse in a so-called atomic set. We provide general recovery results stating that a convex programming can stably and robustly recover signals if the null space of the sensing map satisfies certain properties. Moreover, we argue that such null space property can be satisfied with high probability if each measurement is sub-Gaussian even when the number of measurements are very few. Some new results for recovering signals sparse in a frame, and recovering low rank matrices are also derived as a result.

Achieving Robust Compressive Sensing Seismic Acquisition with a Two-Step Sampling Approach

The compressive sensing (CS) framework offers a cost-effective alternative to dense alias-free sampling. Designing seismic layouts based on the CS technique imposes the use of specific sampling patterns in addition to the logistical and geophysical requirements. We propose a two-step design process for generating CS-based schemes suitable for seismic applications. During the first step, uniform random sampling is used to generate a random scheme, which is supported theoretically by the restricted isometry property. Following that, designated samples are added to the random scheme to control the maximum distance between adjacent sources (or receivers). The null space property theoretically justifies the additional samples of the second step. Our sampling method generates sampling patterns with a CS theoretical background, controlled distance between adjacent samples, and a flexible number of active and omitted samples. The robustness of two-step sampling schemes for reallocated samples is investigated and CS reconstruction tests are performed. In addition, using this approach, a CS-based 3D seismic survey is designed, and the distributions of traces in fold maps and rose diagrams are analyzed. It is shown that the two-step scheme is suitable for CS-based seismic surveys and field applications.

Cooperative learning event‐triggered control for discrete‐time nonlinear multi‐agent systems by internal and external interaction topology

AbstractThis paper investigates distributed cooperative learning event‐triggered control for discrete‐time strict‐feedback nonlinear multi‐agent systems with an identical system structure and different recurrent reference orbits. A clever system decomposition method is firstly proposed to divide every ‐order agent system into first‐order subsystems, which makes it possible to solve the tough problem that every estimated neural weight cannot converge to an unique ideal value based on the existing control scheme. By constructing two interaction typologies, a novel distributed cooperative weight updating law is designed by the introduction of the internal interaction terms between subsystems of every agent, the external interaction terms between agents, and the triggering mechanism between agents. With the combination of the graph multiplication operation, the consensus theory and the matrix null space property, the proposed cooperative event‐triggered control scheme guarantees that every agent can track their corresponding recurrent reference trajectories. And further the exponential convergence of all agents' neural estimated weights to a close vicinity around their mutual and unique ideal weights is verified under the condition that the directed interaction topology between agents is strongly connected and balanced. Such a weight convergence makes the proposed scheme has some significant advantages including the small data storage space, the convenient knowledge reuse, the good generalization ability, and the low communication burden. By reinvoking the saved constant weights, a static learning controller is presented for the high‐performance control of similar diversified tracking tasks. Simulation studies and some comparisons are given to show the advantages and effectiveness of the presented scheme.

The null space property of the weighted ℓr − ℓ1 minimization

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This paper studies NSP of the weighted [Formula: see text] ([Formula: see text]) minimization. Several versions of NSP of the weighted [Formula: see text] minimization including the weighted [Formula: see text] NSP, the weighted [Formula: see text] stable NSP, the weighted [Formula: see text] robust NSP and the [Formula: see text] weighted [Formula: see text] robust NSP for [Formula: see text], are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted [Formula: see text] minimization are established. Furthermore, we show that to some extent, the weighted [Formula: see text] stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022).

Null space property for compressed sensing with frames via l_q (0 < q ≤ 1) synthesis

Null space property for compressed sensing with frames via l_q (0 < q ≤ 1) synthesis

Distributed Resilient Observer: Blended Dynamics Theory Meets ℓ1-Minimization Approach

This paper presents a distributed resilient observer for continuous-time linear time-invariant plants that remains functional even under sensor attacks. The proposed method aims to determine the estimation outcome that matches the majority of sensor measurements, which is formulated as an ℓ1-minimization problem considering all the observable components of each sensor measurement. A distributed observer based on the blended dynamics theory is then proposed to solve the ℓ1-minimization problem in a distributed manner. As a result, the distributed resilient estimation is enabled for a broader class of systems compared to previous works. The design procedure is constructive with parameters obtained from a specified condition that is equivalent to the well-known null-space property.

An analysis of noise folding for low-rank matrix recovery

Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with double noise model is larger by a factor to [Formula: see text], where [Formula: see text] are the dimensions of the matrix and [Formula: see text] is the number of measurements. Additionally, this paper discusses the reconstruction of low-rank matrices in the setting, presents sufficient conditions based on the associating null space property to guarantee the robust recovery and obtains the number of measurements. Furthermore, for the non-Gaussian noise scenario, we further explore it and give the corresponding result. The simulation experiments conducted, on the one hand show effect of noise variance on recovery performance, on the other hand demonstrate the verifiability of the proposed model.

On the Support Recovery of Jointly Sparse Gaussian Sources via Sparse Bayesian Learning

In this work, we provide non-asymptotic, probabilistic guarantees for successful recovery of the common nonzero support of jointly sparse Gaussian sources in the multiple measurement vector (MMV) problem. The support recovery problem is formulated as the marginalized maximum likelihood (or type-II ML) estimation of the variance hyperparameters of a joint sparsity inducing Gaussian prior on the source signals. We derive conditions under which the resulting nonconvex constrained optimization perfectly recovers the nonzero support of a joint-sparse Gaussian source ensemble with arbitrarily high probability. The support error probability decays exponentially with the number of MMVs at a rate that depends on the smallest restricted singular value and the nonnegative null space property of the self Khatri-Rao product of the sensing matrix. Our analysis confirms that nonzero supports of size as high as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(m^{2})$ </tex-math></inline-formula> are recoverable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> measurements per sparse vector. Our derived sufficient conditions for support consistency of the proposed constrained type-II ML solution also guarantee the support consistency of any global solution of the multiple sparse Bayesian learning (M-SBL) optimization whose nonzero coefficients lie inside a bounded interval. For the case of noiseless measurements, we further show that a single MMV is sufficient for perfect recovery of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -sparse support by M-SBL, provided all subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k + 1$ </tex-math></inline-formula> columns of the sensing matrix are linearly independent.

Exact recovery of sparse signals with side information

Compressed sensing has captured considerable attention of researchers in the past decades. In this paper, with the aid of the powerful null space property, some deterministic recovery conditions are established for the previous ell _{1}–ell _{1} method and the ell _{1}–ell _{2} method to guarantee the exact sparse recovery when the side information of the desired signal is available. These obtained results provide a useful and necessary complement to the previous investigation of the ell _{1}–ell _{1} and ell _{1}–ell _{2} methods that are based on the statistical analysis. Moreover, one of our theoretical findings also shows that the sharp conditions previously established for the classical ell _{1} method remain suitable for the ell _{1}–ell _{1} method to guarantee the exact sparse recovery. Numerical experiments on both the synthetic signals and the real-world images are also carried out to further test the recovery performance of the above two methods.

Robust recovery of a kind of weighted l1-minimization without noise level

This paper studies a kind of weighted [Formula: see text]-minimization that is motivated by function interpolation. Combining with the weighted robust null space property, we first propose a new sufficient condition for robust recovery via the weighted [Formula: see text]-minimization when the measurements are corrupted by arbitrary noise without requiring the proper estimation of noise level. Second, we investigate the instance optimality of the weighted [Formula: see text]-minimization decoder according to the weighted quotient property and the weighted restricted isometry property (RIP). In addition, to give a better error estimation in a general problem to recover noisy compressible signals, we improve the [Formula: see text]-RIP constant [Formula: see text] from [Formula: see text] to [Formula: see text] by using the weighted sparsity and Stechkin-type estimate. Our results show that the weighted [Formula: see text]-minimization remains not only stable but also robust to reconstruct signals with noisy observations.

Stable recovery of weighted sparse signals from phaseless measurements via weighted l1 minimization

The goal of phaseless compressed sensing is to recover an unknown sparse or approximately sparse signal from the magnitude of its measurements. However, it does not take advantage of any support information of the original signal. Therefore, our main contribution in this paper is to extend the theoretical framework for phaseless compressed sensing to incorporate with prior knowledge of the support structure of the signal. Specifically, we investigate two conditions that guarantee stable recovery of a weighted k‐sparse signal via weighted l1 minimization without any phase information. We first prove that the weighted null space property (WNSP) is a sufficient and necessary condition for the success of weighted l1 minimization for weighted k‐sparse phase retrievable. Moreover, we show that if a measurement matrix satisfies the strong weighted restricted isometry property (SWRIP), then the original signal can be stably recovered from the phaseless measurements.

A necessary and sufficient condition for sparse vector recovery via ℓ1 − ℓ2 minimization

In this paper, we focus on ℓ 1 − ℓ 2 minimization model, i.e., investigating the nonconvex model: min ⁡ ‖ x ‖ 1 − ‖ x ‖ 2 s.t. A x = y and provide a null space property of the measurement matrix A such that a vector x can be recovered from Ax via ℓ 1 − ℓ 2 minimization. The ℓ 1 − ℓ 2 minimization model was first proposed by E.Esser, et al (2013) [8] . As a nonconvex model, it is well known that global minimizer and local minimizer are usually inconsistent. In this paper, we present a necessary and sufficient condition for the measurement matrix A such that (1) a vector x can be recovered from Ax via ℓ 1 − ℓ 2 local minimization ( Theorem 4 ); (2) any k -sparse vector x can be recovered from Ax via ℓ 1 − ℓ 2 local minimization ( Theorem 5 ); (3) any k -sparse vector x can be recovered from Ax via ℓ 1 − ℓ 2 global minimization ( Theorem 6 ).