Restricted Isometry Property Condition Research Articles

Seismic wavelet shape-oriented reflectivity inversion method

Abstract Reflectivity inversion plays a pivotal role in reservoir prediction. Conventional sparse-spike deconvolution assumes that the reflectivity (reflection coefficient) is sparse, which is solved based on the l1 norm. However, the restricted isometry property (RIP) of wavelet matrix and seismic effective bandwidth limits the accuracy of the sparse-spike reflectivity inversion. Consequently, we investigate the connection between seismic amplitude shape and reflectivity. When the reflectivity contains more non-zero values, the wavelet bandwidth within the effective seismic data bandwidth approaches a limit corresponding to the Sinc wavelet, where the main-lobe amplitude closely approximates the reflectivity. Conversely, when the reflectivity has fewer non-zero values, a wavelet with a smaller sidelobe provides a more accurate approximation of the reflectivity. In this paper, we propose a high-resolution inversion optimization method based on joint l2 norm and l1 norm constraints. By parameter tuning, we construct the Sinc wavelet or the wavelet with a weak-sidelobe corresponding to the seismic spectrum. Subsequently, we determine the extremum to approximate the reflectivity. To mitigate the RIP condition's constraints, we employ the l2 norm to balance the l1 norm (joint constraint) by introducing l2 norm with low-pass filtering characteristics. This approach yields more accurate reflectivity estimates. By taking the extremum, this approach yields more accurate reflectivity estimates. The synthetic test demonstrates that our method achieves better reflectivity inversion accuracy compared to sparse-spike inversion with l1–l2 norm constraint. Furthermore, field tests indicate that the proposed reflectivity inversion method not only can better match the well curve, but also exhibits excellent resolution.

Sparse learning model with embedded RIP conditions for turbulence super-resolution reconstruction

In practical engineering scenarios, constraints arising from sensor placement, quantity, and the limitations of current testing technologies often lead to turbulence data characterized by low resolution and irregular structures. Turbulence super-resolution reconstruction is crucial for extracting finer details from irregularly structured, low-resolution measurement data, thereby facilitating comprehensive flow field analyses. This article introduces a sparse learning model (Embedding Restricted Isometry Property Autoencoder (RIP-AE)) for achieving super-resolution reconstruction of flow fields by obtaining compressed representations and sparse transform domain information. In this model, we embed the Restricted Isometry Property (RIP) condition to ensure the effectiveness of compressed representations and sparse transform domains, thereby enhancing the super-resolution reconstruction accuracy of the model. To assess the performance of the RIP-AE model, we utilized data sets of flow around a cylinder at different Reynolds numbers generated by unsteady Reynolds-Averaged Navier–Stokes simulations. We sequentially validate the effectiveness of the RIP condition at different Reynolds numbers (ranging from 1,000 to 500,000) and compare the reconstruction results of the RIP-AE model with those of the downsampled skip-connection/multi-scale (DSC/MS) and MS-AE models under different sampling ratios. The results indicate that the RIP-AE model excels in terms of L2 relative error and demonstrates the capability to achieve high-precision flow field reconstruction even under high sampling ratios.

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.

Enhanced total variation minimization for stable image reconstruction

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature on image enhancement with the TV regularization, and show that the resulting enhanced TV minimization model is particularly effective for reducing the loss of contrast. The main purpose of this paper is to establish stable reconstruction guarantees for the enhanced TV model from noisy subsampled measurements with two sampling strategies, non-adaptive sampling for general linear measurements and variable-density sampling for Fourier measurements. In particular, under some weaker restricted isometry property conditions, the enhanced TV minimization model is shown to have tighter reconstruction error bounds than various TV-based models for the scenario where the level of noise is significant and the amount of measurements is limited. The advantages of the enhanced TV model are also numerically validated by preliminary experiments on the reconstruction of some synthetic, natural, and medical images.

Multilevel sampling based sparse polynomial expansion for efficient uncertainty analysis of mechanical system

The polynomial chaos is widely used in uncertainty analysis of the mechanical system, in which, the samplings hugely affect the efficiency and accuracy. In this paper, a multilevel sampling based sparse polynomial chaos approach is presented for efficient uncertain analysis of the mechanical system, in which, 1) the dynamic responses are approximated with sparse polynomial chaos expansion which enables the sampling number to be much less than that of the polynomial terms, and the number of deterministic systems would be reduced, 2) the multilevel sampling based resampling strategy is applied to weaken the Restricted Isometry Property condition for sparse polynomial chaos expansion based equations solving, and in addition, 3) such multilevel sampling based on evaluating system coherence with generalized coordinates acceleration is used to reduce the motion equations with deterministic parameters for obtaining the dynamic responses accurately and efficiently. Finally, the presented approach is validated with three examples. The results are shown that the sparse polynomial chaos can be utilized to calculate the mean and standard deviation with 1E-5 and 1E-2 accuracy. The sparse polynomial chaos based on multilevel sampling can be used to reduce the sampling size by 25% compared with the Latin Hypercube sampling. Compared with coherence optimal sampling, the convergence of the proposed sampling method is more stable as the sampling size is increased. Lastly, multilevel sampling also can be used to reduce the deterministic motion equations by 20% to obtain 1E-5 accuracy dynamic responses compared with the Latin Hypercube sampling.

An efficient semismooth Newton method for adaptive sparse signal recovery problems

We know that compressive sensing can establish stable sparse recovery results from highly undersampled data under a restricted isometry property condition. In reality, however, numerous problems are coherent, and vast majority conventional methods might work not so well. Recently, it was shown that using the difference between - and -norm as a regularization always has superior performance. In this paper, we consider an adaptive - model where the -norm with measures the data fidelity and the -term measures the sparsity. This proposed model has the ability to deal with different types of noises and extract the sparse property even under high coherent condition. We use a proximal majorization-minimization technique to handle the non-convex regularization term and then employ a semismooth Newton method to solve the corresponding convex relaxation subproblem. We prove that the sequence generated by the semismooth Newton method admits fast local convergence rate to the subproblem under some technical assumptions. Finally, we do some numerical experiments to demonstrate the superiority of the proposed model and the progressiveness of the proposed algorithm.

Some results on OMP algorithm for MMV problem

In this paper, the robustness of the orthogonal matching pursuit (OMP) algorithm for multiple measurement vector (MMV) problem under the restricted isometry property (RIP) is presented. Moreover, it is shown that the OMP algorithm can exactly recover K‐row sparse matrices in K iterations under the RIP condition we have presented. Furthermore, the condition is sharp.

An optimal condition of robust low-rank matrices recovery

In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient conditions $\delta_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee the robust reconstruction $(z\neq0)$ or exact reconstruction $(z=0)$ of all rank $r$ matrices $X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via nuclear norm minimization. Furthermore, we not only show that when $t=1$, the upper bound of $\delta_r<1/3$ is the same as the result of Cai and Zhang \cite{Cai and Zhang}, but also demonstrate that the gained upper bounds concerning the recovery error are better. Moreover, we prove that the restricted isometry property condition is sharp. Besides, the numerical experiments are conducted to reveal the nuclear norm minimization method is stable and robust for the recovery of low-rank matrix.

Optimal Restricted Isometry Condition of Normalized Sampling Matrices for Exact Sparse Recovery With Orthogonal Least Squares

In this paper, we analyze the performance guarantee of the orthogonal least squares (OLS) algorithm expressed in terms of the restricted isometry property (RIP). We show the optimality of the proposed guarantee by showing that there exists an example for which OLS fails to work when the proposed condition is violated. In addition, we show that if the columns of a sampling matrix are $\ell _{2}$ -normalized, then the proposed condition is an optimal recovery guarantee for the orthogonal matching pursuit (OMP) algorithm. We also establish a nearly optimal performance guarantee of OLS in the more general case where a sampling matrix might not have unit $\ell _{2}$ -norm columns. Moreover, we analyze the performance of OLS in the noisy case. Our result demonstrates that under a suitable constraint on the minimum magnitude of nonzero elements in an input signal, the proposed RIP condition ensures OLS to identify the support exactly. By providing a necessary condition on the minimum magnitude, we show that the proposed minimum magnitude condition is nearly optimal.

Joint Sparse Recovery Using Signal Space Matching Pursuit

In this paper, we put forth a new joint sparse recovery algorithm called signal space matching pursuit (SSMP). The key idea of the proposed SSMP algorithm is to sequentially investigate the support of jointly sparse vectors to minimize the subspace distance to the residual space. Our performance guarantee analysis indicates that SSMP accurately reconstructs any row K -sparse matrix of rank r in the full row rank scenario if the sampling matrix $\mathbf {A}$ satisfies $\mathop{\mathrm{krank}}\limits (\mathbf {A}) \ge {K}+1$ , which meets the fundamental minimum requirement on $\mathbf {A}$ to ensure exact recovery. We also show that SSMP guarantees exact reconstruction in at most ${K}-{r}+\lceil \frac {{r}}{{L}} \rceil $ iterations, provided that $\mathbf {A}$ satisfies the restricted isometry property (RIP) of order ${L}({K}-{r})+{r}+1$ with $\delta _{{L}({K}-{r})+{r}+1} , where L is the number of indices chosen in each iteration. This implies that the requirement on the RIP constant becomes less restrictive when r increases. Such behavior seems to be natural but has not been reported for most of conventional methods. We also show that if ${r}=1$ , then by running more than K iterations, the performance guarantee of SSMP can be improved to $\delta _{\lfloor 7.8{K} \rfloor } \le 0.155$ . Furthermore, we show that under a suitable RIP condition, the reconstruction error of SSMP is upper bounded by a constant multiple of the noise power, which demonstrates the robustness of SSMP to measurement noise. Finally, from extensive numerical experiments, we show that SSMP outperforms conventional joint sparse recovery algorithms both in noiseless and noisy scenarios.

Fast Thresholding Algorithms with Feedbacks and Partially Known Support for Compressed Sensing

Some works in modified compressive sensing (CS) show that reconstruction of sparse signals can obtain better results than traditional CS using the partially known support. In this paper, we extend the idea of these works to the null space tuning algorithm with hard thresholding, feedbacks ([Formula: see text]) and derive sufficient conditions for robust sparse signal recovery. The theoretical analysis shows that including prior information of partially known support relaxes the preconditioned restricted isometry property condition comparing with the [Formula: see text]. Numerical experiments demonstrate that the modification improves the performance of the NST+HT+FB, thereby requiring fewer samples to obtain an approximate reconstruction. Meanwhile, a systemic comparison with different methods based on partially known support is shown.

Uniform RIP Conditions for Recovery of Sparse Signals by $\ell _p\,(0&lt; p\leq 1)$ Minimization

Compressed sensing in both noiseless, and noisy cases is considered in this article, and uniform restricted isometry property (RIP) conditions for sparse signal recovery are established via $\ell _p\,(0 minimization. It is shown that if the measurement matrix satisfies the sharp condition $\Phi (p,t)>0$ for any given constant $t>1$ , where $\Phi (p,t)$ concerning the restricted isometry constants $\delta _{tk}$ , and $\delta _{2(t-1)k}$ is specified in the context, then all $k$ -sparse signals can be exactly recovered by the constrained $\ell _p$ minimization. This uniform RIP framework with general $p$ , and $t$ includes three state-of-the-art results concerning $p=1$ , $t=2$ , and $t\in [\frac{4}{2+p},2]$ as special cases. Utilizing higher-order RIP conditions can result in a milder sufficient condition for sparse recovery. For $t\geq 2$ , the RIP condition $\delta _{tk} , where the upper bound $\delta (p,t)$ is defined in the context, is shown to be sufficient to guarantee both the exact recovery of all $k$ -sparse signals in the noiseless case, and the stable recovery of approximately $k$ -sparse signals in noisy cases. Moreover, we establish a threshold of the restricted isometry constant $\delta _{tk}$ where the failure of $\ell _p$ sparse recovery will occur.

2D compressed learning: support matrix machine with bilinear random projections

Support matrix machine (SMM) is an efficient matrix classification method that can leverage the structure information within the matrix to improve the classification performance. However, its computational and storage costs are still expensive for high-dimensional data. To address these problems, in this paper, we consider a 2D compressed learning paradigm to learn the SMM classifier in some compressed data domain. Specifically, we use the Kronecker compressed sensing (KCS) to obtain the compressive measurements and learn the SMM classifier. We show that the Kronecker product measurement matrices used by KCS satisfies the restricted isometry property (RIP), which is a property to ensure the learnability of the compressed data. We further give a lower bound on the number of measurements required for KCS. Though this lower bound shows that KCS requires more measurements than the regular CS to satisfy the same RIP condition, KCS itself still enjoys lower computational and storage complexities. Then, using the RIP condition, we verify that the learned SMM classifier in the compressed domain can perform almost as well as the best linear classifier in the original uncompressed domain. Finally, our experimental results also demonstrate the feasibility of 2D compressed learning.

Efficiency of orthogonal super greedy algorithm under the restricted isometry property

We investigate the efficiency of orthogonal super greedy algorithm (OSGA) for sparse recovery and approximation under the restricted isometry property (RIP). We first show that under the RIP conditions of the measurement matrix Φ and the minimum magnitude of the nonzero coordinates of the signal, for l_{2} bounded or l_{infty } bounded noise vector e, with explicit stopping rules, OSGA can recover the support of an arbitrary K-sparse signal x from y=varPhi x +e in at most K steps. Then, we investigate the error performance of OSGA in m term approximation with regards to dictionaries satisfying the RIP in a separable Hilbert space. We establish a Lebesgue-type inequality for OSGA. Based on this inequality, we obtain the optimal rate of convergence for the sparse class induced by such dictionaries.

Sparse Recovery via Partial Regularization: Models, Theory, and Algorithms

In the context of sparse recovery, it is known that most of the existing regularizers such as ℓ1 suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We show that every local minimizer of these models is substantially sparse or the magnitude of all of its nonzero entries is above a uniform constant depending only on the data of the linear system. Moreover, for a class of partial regularizers, any global minimizer of these models is a sparsest solution to the linear system. We also establish some sufficient conditions for local or global recovery of the sparsest solution to the linear system, among which one of the conditions is weaker than the best-known restricted isometry property condition for sparse recovery by ℓ1. In addition, a first-order augmented Lagrangian (FAL) method is proposed for solving these models, in which each subproblem is solved by a nonmonotone proximal gradient (NPG) method. Despite the complication of the partial regularizers, we show that each proximal subproblem in NPG can be solved as a certain number of one-dimensional optimization problems, which usually have a closed-form solution. We also show that any accumulation point of the sequence generated by FAL is a first-order stationary point of the models. Numerical results on compressed sensing and sparse logistic regression demonstrate that the proposed models substantially outperform the widely used ones in the literature in terms of solution quality.

Multiobjective Sparse Non-Negative Matrix Factorization.

Non-negative matrix factorization (NMF) is becoming increasingly popular in many research fields due to its particular properties of semantic interpretability and part-based representation. Sparseness constraints are usually imposed on the NMF problems in order to achieve potential features and sparse representation. These constrained NMF problems are usually reformulated as regularization models to solve conveniently. However, the regularization parameters in the regularization model are difficult to tune and the frequently used sparse-inducing terms in the regularization model generally have bias effects on the induced matrix and need an extra restricted isometry property (RIP). This paper proposes a multiobjective sparse NMF paradigm which refrains from the regularization parameter issues, bias effects, and the RIP condition. A novel multiobjective memetic algorithm is also proposed to generate a set of solutions with diverse sparsity and high factorization accuracy. A masked projected gradient local search scheme is specially designed to accelerate the convergence rate. In addition, a priori knowledge is also integrated in the algorithm to reduce the computational time in discovering our interested region in the objective space. The experimental results show that the proposed paradigm has better performance than some regularization algorithms in producing solutions with different degrees of sparsity as well as high factorization accuracy, which are favorable for making the final decisions.

On uniqueness of sparse signal recovery

Abstract A basic issue of sparse signal recovery (SSR) is to explore the condition of the uniqueness with regard to the solution of the relevant optimization framework. However, the standard uniqueness conditions, such as spark condition, NSP (null space property), RIP (restricted isometry property) and mutual coherence condition, are with respect to any sparse signal with the same sparsity. Therefore, these four conditions require certain structural or metric properties of all possible sub-matrices from the measurement matrix corresponding to the possible support indices and are quite restrictive for a given sparse signal with its support indices fixed. This work mainly considers the uniqueness issue of SSR. With the extra information of the support indices, the requirements of the measurement matrix for guaranteeing the uniqueness are released. Theoretical analysis has been performed on the uniqueness for l0-norm and l1-norm frameworks, in which loosed conditions are further validated by constructed examples. These discoveries can explain the phenomena that the occasional success of SSR in numerical simulation occurs when the above four conditions cannot be satisfied. Besides, it is analyzed that the spark condition and NSP condition are the minimal requirements for the unique recovery of the standard l0-norm framework and the l1-norm framework respectively.

Smoothing inertial projection neural network for minimization [formula omitted] in sparse signal reconstruction

In this paper, we investigate a more general sparse signal recovery minimization model and a smoothing neural network optimal method for compress sensing problem, where the objective function is a Lp−q minimization model which includes nonsmooth, nonconvex, and non-Lipschitz quasi-norm Lp norms 1≥p>0 and nonsmooth Lq norms 2≥p>1, and its feasible set is a closed convex subset of Rn. Firstly, under the restricted isometry property (RIP) condition, the uniqueness of solution for the minimization model with a given sparsity s is obtained through the theoretical analysis. With a mild condition, we get that the larger of the q, the more effective of the sparse recovery model under sensing matrix satisfies RIP conditions at fixed p. Secondly, using a smoothing approximate method, we propose the smoothing inertial projection neural network (SIPNN) algorithm for solving the proposed general model. Under certain conditions, the proposed algorithm can converge to a stationary point. Finally, convergence behavior and successful recover performance experiments and a comparison experiment confirm the effectiveness of the proposed SIPNN algorithm.

AKRON: An algorithm for approximating sparse kernel reconstruction

Exact reconstruction of a sparse signal for an under-determined linear system using the ℓ0-measure is, in general, an NP-hard problem. The most popular approach is to relax the ℓ0-optimization problem to an ℓ1-approximation. However, the strength of this convex approximation relies upon rigid properties on the system, which are not verifiable in practice. Greedy algorithms have been proposed in the past to speed up the optimization of the ℓ1 problem, but their computational efficiency comes at the expense of a larger error. In an effort to control error and complexity, this paper goes beyond the ℓ1-approximation by growing neighborhoods of the ℓ1-solution that moves towards the optimal solution. The size of the neighborhood is tunable depending on the computational resources. The proposed algorithm, termed Approximate Kernel RecONstruction (AKRON), yields significantly smaller errors than current greedy methods with a controllable computational cost. By construction, the error of AKRON is smaller than or to equal the ℓ1-solution. AKRON enjoys all the error bounds of ℓ1 under the restricted isometry property condition. We benchmarked AKRON on simulated data from several under-determined systems, and the results show that AKRON can significantly improve the reconstruction error with slightly more computational cost than solving the ℓ1 problem directly.

Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

This paper establishes new sufficient conditions on the restricted isometry property (RIP) for compressed sensing with coherent tight frames. One of our main results shows that the RIP (adapted to D) condition δk+θk,k&lt;1 guarantees the stable recovery of all signals that are nearly k-sparse in terms of a coherent tight frame D via the l1-analysis method, which improves the existing ones in the literature.