Smooth Approximation Research Articles

Improved local smoothing estimate for the wave equation in higher dimensions

In this paper, we establish the sharp k-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions n≥3. As a byproduct, we also generalize the restriction estimates of Ou–Wang [21] to a broader class of phase functions.

Clubfoot from past to the present: A bibliometric analysis with global productivity and research trends.

Clubfoot, a common congenital abnormality, affects the lower extremities; however, the literature search revealed no bibliometric research on this subject. Thus, we aimed to holistically analyze scientific articles and reveal global productivity and trend issues. This study statistically analyzed 1417 published articles on clubfoot (1980-2021) from the Web of Science database. Bibliometric network visualization maps were created to reveal trend topics, citation analysis, and cross-country collaborations. The analysis was conducted using Spearman correlation analysis. An exponential smoothing estimator was used to predict article productivity. The United States of America (433, 30.5%), the United Kingdom (166, 11.7%), and India (107, 7.5%) are the top 3 countries contributing to the literature. The Journal of Pediatric Orthopedics (220 articles), the Journal of Pediatric Orthopedics-Part B (147 articles), and Clinical Orthopedics and Related Research (69 articles) are the top 3 most productive journals. Dobbs MB (34 articles) is the most active author, and Shriners Hospital Children (44 articles) is the most active institution. Bibliometric analysis revealed that recently studied trend topics included Pirani score, Dimeglio score, Ponseti method, Ponseti casting, tenotomy, recurrence, neglected, tendon transfer, bracing, gait, risk factors, pedobarography, complex clubfoot, and polymorphism. The most studied subjects included Ponseti technique, treatment/casting, recurrent/relapsed clubfoot, Pirani score, pediatrics/children, foot deformities, surgery, ultrasound, Achilles tendon/tenotomy, gait analysis, casting, outcomes, neglected clubfoot, and tenotomy. Research leadership was determined in the western and European countries and Canada in studies and scientific collaborations on clubfoot; its impact was remarkable in India, China, and Turkey.

Smooth magnetotelluric impedance estimation by optimization

In urban and industrialised areas, magnetotelluric (MT) recordings often contain a significant amount of high-power near-field noise which, constituting non-Gaussian noise for most of the recording length, may lead to the failure of the common transfer functions (TF) estimation methods. The most evident consequence of this is the appearance of sharp points in the TF curves versus frequency, deteriorating their inherent smoothness. The TF's smoothness is in fact one of the main criteria adopted by MT practitioners to evaluate the data quality and effectiveness of the estimation procedure; when not reached by the estimation, it may be a-posteriori introduced by splines or filtering procedures, but these approaches lack physical soundness. We propose a new heuristic algorithm aimed at obtaining the maximum TF versus frequency smoothness, through an optimization scheme rejecting EM events in frequency-domain. The algorithm searches for frequency-dependent power thresholds to be applied to the events, rejecting those making the TF sharp. The TF's smoothness is the objective function to be minimized, and the model space is constituted by the infinite set of threshold vectors for each analysed frequency. After the process, the Ordinaly Least Squares (OLS) residual distribution is much closer to a Rayleigh one, allowing for an unbiased estimation. We applied the procedure both to synthetic and real MT data. Measured data has been collected for a geothermal exploration project in the East Africa Rift, where strong coherent noise affected the recordings. The combination of the technique with the remote-reference technique effectively reduced the bias due to the artificial signal. The physical consistency of the resulting TF has been tested by dispersion relationships.

JUTAR: Joint User-Association, Task-Partition, and Resource-Allocation Algorithm for MEC Networks.

Mobile edge computing (MEC) is a promising technique to support the emerging delay-sensitive and compute-intensive applications for user equipment (UE) by means of computation offloading. However, designing a computation offloading algorithm for the MEC network to meet the restrictive requirements towards system latency and energy consumption remains challenging. In this paper, we propose a joint user-association, task-partition, and resource-allocation (JUTAR) algorithm to solve the computation offloading problem. In particular, we first build an optimization function for the computation offloading problem. Then, we utilize the user association and smooth approximation to simplify the objective function. Finally, we employ the particle swarm algorithm (PSA) to find the optimal solution. The proposed JUTAR algorithm achieves a better system performance compared with the state-of-the-art (SOA) computation offloading algorithm due to the joint optimization of the user association, task partition, and resource allocation for computation offloading. Numerical results show that, compared with the SOA algorithm, the proposed JUTAR achieves about 21% system performance gain in the MEC network with 100 pieces of UE.

A modified sequential quadratic programming method for sparse signal recovery problems

We propose a modified sequential quadratic programming method for solving the sparse signal recovery problem. We start by going through the well-known smoothed-ℓ0 technique and provide a smooth approximation of the objective function. Then, a variant of the sequential quadratic programming method equipped with a new approach for solving subproblems is proposed. We investigate the global convergence of the method in detail. In comparison to several well-known algorithms, simulation results demonstrate the promising performance of the proposed method.

Robust feature matching via progressive smoothness consensus

Feature matching is a long-standing fundamental and critical problem in computer vision and photogrammetry. The indirect matching strategy has become a popular choice because of its high precision and generality, but it finds only a limited number of correct matches, and the mismatch removal phase does not utilize the critical feature descriptors. To this end, this paper proposes a novel and effective feature matching method, named Progressive Smoothness Consensus (PSC). Our PSC designs an objective function to directly construct correct matches from two feature point sets. To optimize the objective, we introduce a stepwise strategy, where a small but reliable match set with the smooth function is used as initialization, and then the correct match set is iteratively enlarged and optimized by match expansion and smooth function estimation, respectively. In addition, the local geometric constraint is added to the compact representation with a Fourier basis, thus improving the estimation precision. We perform the match expansion as a Bayesian formulation to exploit both the spatial distribution and feature description information, thus finding feasible matches to expand the match set. Extensive experiments on feature matching, homography & fundamental matrix estimation, and image registration are conducted, which demonstrate the advantages of our PSC against state-of-the-art methods in terms of generality and effectiveness. Our code is publicly available at https://github.com/XiaYifan1999/Robust-feature-matching-via-Progressive-Smoothness-Consensus.

Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.

Smooth approximation of mappings with rank of the derivative at most 1

It was conjectured that if $$f\in C^1({\mathbb {R}}^n,{\mathbb {R}}^n)$$ satisfies $$\mathrm{rank\,}Df\le m<n$$ everywhere in $${\mathbb {R}}^n$$ , then f can be uniformly approximated by $$C^\infty $$ -mappings g satisfying $$\mathrm{rank\,}Dg\le m$$ everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when $$m=1$$ . More precisely, if $$m=1$$ , our result yields an almost-uniform approximation of locally Lipschitz mappings $$f:\Omega \rightarrow {\mathbb {R}}^n$$ , satisfying $$\mathrm{rank\,}Df\le 1$$ a.e., by $$C^\infty $$ -mappings g with $$\mathrm{rank\,}Dg\le 1$$ , provided $$\Omega \subset {\mathbb {R}}^n$$ is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ( $${\mathbb {R}}$$ -trees).

Reducing the Complexity of Two Classes of Optimization Problems by Inexact Accelerated Proximal Gradient Method

We propose a double-loop inexact accelerated proximal gradient (APG) method for a strongly convex composite optimization problem with two smooth components of different smoothness constants and computational costs. Compared to APG, the inexact APG can reduce the time complexity for finding a near-stationary point when one smooth component has higher computational cost but a smaller smoothness constant than the other. The strongly convex composite optimization problem with this property arises from subproblems of a regularized augmented Lagrangian method for affine-constrained composite convex optimization and also from the smooth approximation for bilinear saddle-point structured nonsmooth convex optimization. We show that the inexact APG method can be applied to these two problems and reduce the time complexity for finding a near-stationary solution. Numerical experiments demonstrate significantly higher efficiency of our methods over an optimal primal-dual first-order method by Hamedani and Aybat [SIAM J. Optim., 31 (2021), pp. 1299–1329] and the gradient sliding method by Lan, Ouyang, and Zhou [arXiv2101.00143, 2021].

(Retracted) Simulating human visual perception behaviors for video quality assessment

Multimedia video quality evaluation system plays an important role in modern Internet applications, such as WebRTC-based video conference. However, video quality may be affected during transmission because of noise or network congestion. Constructing an end-to-end intelligent video transmission monitoring system is of great significance. We design a system based on no-reference quality assessment, which can be used to evaluate the video-based object quality evaluation. More specifically, we first leverage Gabor filter for video segmentation. Then we conduct image smoothness estimation to measure the correlation between two adjacent frames. Subsequently, we consider two states of video distortion: compression and video transmission. We model these two states, respectively. We compare our VQA algorithm with a set of well-known NR-IQA. The reported VQA accuracies have clearly shown the superiority of our proposed video quality predictor.

Smooth approximation of twisted Kähler-Einstein metrics

Abstract In this article, we prove the existence of smooth approximations of twisted Kähler-Einstein metrics using the variational method.

Orthogonal series method for uncertain nonparametric regression with application to carbon dioxide emissions

Parametric regression is an important branch of statistics that assumes the regression model is known with a mathematical expression, while nonparametric regression has few restrictions on the regression model that is determined by the data. As a tool to explore the relationship between the observations characterized by uncertain variables, uncertain nonparametric regression has not obtained enough attention except for B-spline and local polynomial regression. In order to get a more global and smooth estimate, this paper employs the orthogonal series to approximate the nonparametric regression model. We intend to choose an appropriate number of the orthogonal bases by the leave-one-point-out cross-validation. Then, a new prediction technology is proposed to derive the response variable’s forecast value and confidence interval. Finally, a numerical example and a real data example of carbon dioxide emissions are put forward to demonstrate the effectiveness and accuracy of the method.

Histopolating fractal functions

Fractal interpolation functions that produce smooth and non-smooth approximants constitute an advancement to various classical nonrecursive methods of approximation. However, in fractal approximation, the emphasis so far has been on the investigation of continuous functions interpolating prescribed data sets. In this article, we exclude the two properties – continuity and interpolation – inherent in fractal interpolation, and introduce the concept of fractal histopolation. That is, we study the area-matching properties of fractal functions that are integrable but not necessarily continuous.

An approximate Bayesian approach for estimation of the instantaneous reproduction number under misreported epidemic data.

In epidemic models, the effective reproduction number is of central importance to assess the transmission dynamics of an infectious disease and to orient health intervention strategies. Publicly shared data during an outbreak often suffers from two sources of misreporting (underreporting and delay in reporting) that should not be overlooked when estimating epidemiological parameters. The main statistical challenge in models that intrinsically account for a misreporting process lies in the joint estimation of the time-varying reproduction number and the delay/underreporting parameters. Existing Bayesian approaches typically rely on Markov chain Monte Carlo algorithms that are extremely costly from a computational perspective. We propose a much faster alternative based on Laplacian-P-splines (LPS) that combines Bayesian penalized B-splines for flexible and smooth estimation of the instantaneous reproduction number and Laplace approximations to selected posterior distributions for fast computation. Assuming a known generation interval distribution, the incidence at a given calendar time is governed by the epidemic renewal equationand the delay structure is specified through a composite link framework. Laplace approximations to the conditional posterior of the spline vector are obtained from analytical versions of the gradient and Hessian of the log-likelihood, implying a drastic speed-up in the computation of posterior estimates. Furthermore, the proposed LPS approach can be used to obtain point estimates and approximate credible intervals for the delay and reporting probabilities. Simulation of epidemics with different combinations for the underreporting rate and delay structure (one-day, two-day, and weekend delays) show that the proposed LPS methodology delivers fast and accurate estimates outperforming existing methods that do not take into account underreporting and delay patterns. Finally, LPS is illustrated in two real case studies of epidemicoutbreaks.

From [formula omitted] subgradient to projection: A compact neural network for [formula omitted]-regularized logistic regression

ℓ1 regularization has been used for logistic regression to circumvent the overfitting and use the estimated sparse coefficient for feature selection. However, the challenge of such regularization is that the ℓ1 regularization is not differentiable, making the standard convex optimization algorithm not applicable to this problem. This paper presents a simple projection neural network for ℓ1-regularized logistics regression. In contrast to many available solvers in the literature, the proposed neural network does not require any extra auxiliary variable nor smooth approximation, and its complexity is almost identical to that of the gradient descent for logistic regression without ℓ1 regularization, thanks to the projection operator. We also investigate the convergence of the proposed neural network by using the Lyapunov theory and show that it converges to a solution of the problem with any arbitrary initial value. The proposed neural solution significantly outperforms state-of-the-art methods concerning the execution time and is competitive in terms of accuracy and AUROC.

Selection of the optimal personalized treatment from multiple treatments with right-censored multivariate outcome measures

We propose a novel personalized concept for the optimal treatment selection for a situation where the response is a multivariate vector that could contain right-censored variables such as survival time. The proposed method can be applied with any number of treatments and outcome variables, under a broad set of models. Following a working semiparametric Single Index Model that relates covariates and responses, we first define a patient-specific composite score, constructed from individual covariates. We then estimate conditional means of each response, given the patient score, correspond to each treatment, using a nonparametric smooth estimator. Next, a rank aggregation technique is applied to estimate an ordering of treatments based on ranked lists of treatment performance measures given by conditional means. We handle the right-censored data by incorporating the inverse probability of censoring weighting to the corresponding estimators. An empirical study illustrates the performance of the proposed method in finite sample problems. To show the applicability of the proposed procedure for real data, we also present a data analysis using HIV clinical trial data, that contained a right-censored survival event as one of the endpoints.

Crime in Philadelphia: Bayesian Clustering with Particle Optimization

Accurate estimation of the change in crime over time is a critical first step toward better understanding of public safety in large urban environments. Bayesian hierarchical modeling is a natural way to study spatial variation in urban crime dynamics at the neighborhood level, since it facilitates principled “sharing of information” between spatially adjacent neighborhoods. Typically, however, cities contain many physical and social boundaries that may manifest as spatial discontinuities in crime patterns. In this situation, standard prior choices often yield overly smooth parameter estimates, which can ultimately produce mis-calibrated forecasts. To prevent potential over-smoothing, we introduce a prior that partitions the set of neighborhoods into several clusters and encourages spatial smoothness within each cluster. In terms of model implementation, conventional stochastic search techniques are computationally prohibitive, as they must traverse a combinatorially vast space of partitions. We introduce an ensemble optimization procedure that simultaneously identifies several high probability partitions by solving one optimization problem using a new local search strategy. We then use the identified partitions to estimate crime trends in Philadelphia between 2006 and 2017. On simulated and real data, our proposed method demonstrates good estimation and partition selection performance. Supplementary materials for this article are available online.

Nonlinear wave equations with slowly decaying initial data

New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via ℓ2-decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The results are compared to new well-posedness results in Lp-based Sobolev spaces.

Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation

In the Compressed Sensing (CS) framework, the underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possible solution, which is l0-norm minimization. However, finding an l0 norm solution out of infinitely many possible solutions is NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact that l0 norm penalty can be adequately estimated by l1 norm, which recasts a non-convex minimization problem to a convex problem. However, l1 norm non-differentiable and gradient-based minimization algorithms are not applicable, due to this very reason there is a need to approximate l1 norm by its smooth approximation. Iterative shrinkage algorithms provide an efficient method to numerically minimize l1-regularized least square optimization problem. These algorithms are required to induce sparsity in their solutions to meet the CS recovery requirement. In this research article, we have developed a novel recovery method that uses hyperbolic tangent function to recover undersampled signal/images in CS framework. In our work, l1 norm and soft thresholding are both approximated with the hyperbolic tangent functions. We have also proposed the criteria to tune optimization parameters to get optimal results. The error bounds for the proposed l1 norm approximation are evaluated. To evaluate performance of our proposed method, we have utilized a dataset comprised of 1-D sparse signal, compressively sampled MR image and cardiac cine MRI. The MRI is an important imaging modality for assessing cardiac vascular function. It provides the ejection fraction and cardiac output of the heart. However, this advantage comes at the cost of a slow acquisition process. Hence, it is essential to speed up the acquisition process to take the full benefits of cardiac cine MRI. Numerical results based on performance metrics, such as Structural Similarity (SSIM), Peak Signal to Noise Ratio (PSNR) and Root Mean Square Error (RMSE) show that the proposed tangent hyperbolic based CS recovery offers a much better performance as compared to the traditional Iterative Soft Thresholding (IST) recovery methods.

On the Hamilton–Poisson structure and solitons for the Maxwell–Lorentz equations with spinning particle

We construct the Hamilton–Poisson structure for the Maxwell–Lorentz equations with spinning extended charged particle. We calculate the integral kernel of the corresponding Poisson bracket. The calculation relies on the Hamilton least action principle which is proved using the Lie–Poincaré calculus. The Hamilton–Poisson structure turns out to be degenerate and admits a functional family of Casimir invariants. The structure is applied to the calculation of the solitons corresponding to the uniform motion and rotation of the particle. We calculate effective masses and momenta of inertia of the soltions.As an example, the Poisson structure is applied to the construction of the momentum map corresponding to the symmetry group of translations. All calculations are rigorously justified by the proof of the well-posedness and smooth approximations of solutions. The investigation is inspired by the problems of stability of solitons.