A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

$\ell_1$ mean filtering is a conventional, optimization-based method to estimate the positions of jumps in a piecewise constant signal perturbed by additive noise. In this method, the $\ell_1$ norm penalizes sparsity of the first-order derivative of the signal. Theoretical results, however, show that in some situations, which can occur frequently in practice, even when the jump amplitudes tend to $\infty$, the conventional method identifies false change points. This issue is referred to as stair-casing problem and restricts practical importance of $\ell_1$ mean filtering. In this paper, sparsity is penalized more tightly than the $\ell_1$ norm by exploiting a certain class of nonconvex functions, while the strict convexity of the consequent optimization problem is preserved. This results in a higher performance in detecting change points. To theoretically justify the performance improvements over $\ell_1$ mean filtering, deterministic and stochastic sufficient conditions for exact change point recovery are derived. In particular, theoretical results show that in the stair-casing problem, our approach might be able to exclude the false change points, while $\ell_1$ mean filtering may fail. A number of numerical simulations assist to show superiority of our method over $\ell_1$ mean filtering and another state-of-the-art algorithm that promotes sparsity tighter than the $\ell_1$ norm. Specifically, it is shown that our approach can consistently detect change points when the jump amplitudes become sufficiently large, while the two other competitors cannot.Comment: Submitted to IEEE Transactions on Signal Processin

Recovery of Low-Rank Matrices under Affine Constraints via a Smoothed Rank Function

In this paper, the problem of matrix rank minimization under affine constraints is addressed. The state-of-the-art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem. We propose an algorithm based on a smooth approximation of the rank function, which practically improves recovery limits on the rank of the solution. This approximation leads to a non-convex program; thus, to avoid getting trapped in local solutions, we use the following scheme. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy. On the theoretical side, benefiting from the spherical section property, we will show that the sequence of the solutions of the approximating function converges to the minimum rank solution. On the experimental side, it will be shown that the proposed algorithm, termed SRF standing for Smoothed Rank Function, can recover matrices which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization. Furthermore, it will be demonstrated that, in completing partially observed matrices, the accuracy of SRF is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low-rank matrix.Comment: Accepted in IEEE TSP on December 4th, 201

DOA Estimation in Partially Correlated Noise Using Low-Rank/Sparse Matrix Decomposition

We consider the problem of direction-of-arrival (DOA) estimation in unknown partially correlated noise environments where the noise covariance matrix is sparse. A sparse noise covariance matrix is a common model for a sparse array of sensors consisted of several widely separated subarrays. Since interelement spacing among sensors in a subarray is small, the noise in the subarray is in general spatially correlated, while, due to large distances between subarrays, the noise between them is uncorrelated. Consequently, the noise covariance matrix of such an array has a block diagonal structure which is indeed sparse. Moreover, in an ordinary nonsparse array, because of small distance between adjacent sensors, there is noise coupling between neighboring sensors, whereas one can assume that nonadjacent sensors have spatially uncorrelated noise which makes again the array noise covariance matrix sparse. Utilizing some recently available tools in low-rank/sparse matrix decomposition, matrix completion, and sparse representation, we propose a novel method which can resolve possibly correlated or even coherent sources in the aforementioned partly correlated noise. In particular, when the sources are uncorrelated, our approach involves solving a second-order cone programming (SOCP), and if they are correlated or coherent, one needs to solve a computationally harder convex program. We demonstrate the effectiveness of the proposed algorithm by numerical simulations and comparison to the Cramer-Rao bound (CRB).Comment: in IEEE Sensor Array and Multichannel signal processing workshop (SAM), 201

Successive Concave Sparsity Approximation for Compressed Sensing

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $\ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin

Sewer Pipes Condition Prediction Models: A State-of-the-Art Review

Wastewater infrastructure systems deteriorate over time due to a combination of aging, physical, and chemical factors, among others. Failure of these critical structures cause social, environmental, and economic impacts. To avoid such problems, infrastructure condition assessment methodologies are developing to maintain sewer pipe network at desired condition. However, currently utility managers and other authorities have challenges when addressing appropriate intervals for inspection of sewer pipelines. Frequent inspection of sewer network is not cost-effective due to limited time and high cost of assessment technologies and large inventory of pipes. Therefore, it would be more beneficial to first predict critical sewers most likely to fail and then perform inspection to maximize rehabilitation or renewal projects. Sewer condition prediction models are developed to provide a framework to forecast future condition of pipes and to schedule inspection frequencies. The objective of this study is to present a state-of-the-art review on progress acquired over years in development of statistical condition prediction models for sewer pipes. Published papers for prediction models over a period from 2001 through 2019 are identified. The literature review suggests that deterioration models are capable to predict future condition of sewer pipes and they can be used in industry to improve the inspection timeline and maintenance planning. A comparison between logistic regression models, Markov Chain models, and linear regression models are provided in this paper. Artificial intelligence techniques can further improve higher accuracy and reduce uncertainty in current condition prediction models

Consistent Change Point Detection for Piecewise Constant Signals With Normalized Fused LASSO

We consider the problem of offline change point detection from noisy piecewise constant signals. We propose normalized fused LASSO (FL), an extension of the FL, obtained by normalizing the columns of the sensing matrix of the LASSO equivalent. We analyze the performance of the proposed method, and in particular, we show that it is consistent in detecting change points as the noise variance tends to zero. Numerical experiments support our theoretical findings.QC 20170614</p

Comparison of Diagnostic Markers of Aortic Coarctation in Prenatal and Postnatal Echocardiography

Background: The prenatal diagnosis of coarctation of aorta (CoA) remains a major challenge, as the false-positive diagnosis is fairly common. The purpose of this study was to find some useful prenatal sonographic markers compatible with the postnatal diagnosis of CoA. Methods: The study included fetuses suspected of CoA in the second and third-trimester ultrasound tests. All cases were examined by fetal echocardiography at a single ultrasound clinic between 2019 and 2020. The proportion of right and left ventricular size was assessed and ductal/isthmus diameter ratio was measured. A comparison was drawn between the results of neonates with neonatal CoA and neonates without CoA. Results: Of 20 fetuses with suspected prenatal CoA, 3 (15%) had neonatal CoA. The mean ductal/isthmus ratio was significantly higher in the neonates with CoA (1.96 vs. 1.46; p< 0.001). The diagnostic power of ductal/isthmus ratio to detect CoA with a cut point of 1.53 had a sensitivity and specificity of 100% and 70.6%, respectively, a positive and negative predictive value of 37.5% and 100%, respectively, and an overall accuracy of 75%. Conclusion: The ductal/isthmus ratio diameter and ventricular disproportion are significant sonographic markers for the prenatal diagnosis of CoA, and the ductal/isthmus ratio has high sensitivity and specificity compared to postnatal findings