In this paper, we develop a novel greedy randomized progressive and iterative approximation method for least square fitting (GRLSPIA). By introducing a simpler greedy criterion, the GRLSPIA method adjusts the control points randomly and generates a sequence of curves and surfaces. The limit curve and surface converge to the unique least-norm fitting result in expectation. And the convergence rate of the GRLSPIA method can be smaller with the appropriate parameter. Numerical experiments verify its convergence and show its benefits.
In the paper, we introduce a novel image denoising and segmentation technique using the piecewise smooth relaxed total generalized variation. The relaxed total generalized variational model (RTGV) is a combination of gradient operator and weighted divergence operator, which can improve the computational performance. For the denoising method, the RTGV regularization term and a smooth term are introduced to further improve the restored results. For the image segmentation method, we introduce the RTGV regularized piecewise smooth Mumford–Shah segmentation model instead of total variation regularized, which can deal with image with noise and inhomogeneous intensity effectively. In addition, the augmented Lagrangian algorithm is used to iteratively solve the proposed RTGV method, which produced several closed form solutions. Our algorithm is discussed from several aspects, including theoretical analysis, influence of parameters, and comparisons with several state-of-the-art methods. Numerical experimental results show that our algorithms can yield competitive results both visually and numerical errors with reasonable computational cost. Our image denoising algorithm has the advantage in preserving image features and can effectively prevent the staircase artifacts when compared to several other existing methods. Meanwhile, our segmentation algorithm can obtain results with boundaries more consistent with the human visual system and is shown to be able to achieve multi-region segmentation.
In this note, we consider a general maximal wave operator defined by W_a,tf(x)=∫ _ℝ^ne^i(x·ξ +t(x)|ξ |)a(x,ξ )f(ξ )dξ , where the amplitude a∈ L^∞S^m_ρ and t∈ L^∞ . We prove that this operator is bounded on L^2 provided m<(n-1)ρ -n/2. As a direct application, we obtain the well-known result that the maximal wave operator W^* is bounded from the Sobolev space H^s=W^s,2 to L^2 if s>1/2 . This result is known to be sharp for s>1/2. .
Given a sequence of approximate H-surfaces {u_n}_n∈ℕ from a Riemann surface M to a compact Riemannian manifold N with free boundaries u_n(∂ M) lying on a given supporting submanifold 𝒦⊂ N , admitting uniform energy bound and tension fields uniformly bounded in L^p(M, TN) for p ≥6/5 , we employ the doubling annulus Pohozaev-type identity to establish energy identity and no-neck property during bubbling process near free boundaries for {u_n}_n∈ℕ.
We introduce a novel approach for modeling two-way functional trajectories, relying on Kendall’s tau representation of marginal covariance functions and utilizing the concept of product functional principal component analysis. The developed estimation procedure is intuitive and straightforward to implement. Theoretical results supporting its validity are also established. Numerical simulation studies validate its superior performance compared to recently developed methods. Furthermore, the application of this approach to analyze two-way air pollution trajectories demonstrates its practical superiority.
The prediction of time series is a fundamental problem in big data computing, image processing, computer vision, mathematical statistics, computing economy, and risk management. As a result of its volatility and complexity, time series prediction has been relatively slow in research for decades. Recently, machine learning methods have excelled in time series analysis, as well as advancing time series prediction, while time series prediction is still a difficult task due to the weak interpretability of machine learning and the complexity of time series. From the perspective of data fitting, this paper introduces a new fast and effective method for predicting time series. The method is mainly concerned with dividing the time series into short-, medium-, and long-term data based on the periodicity and trend of the series, which will allow the program to correctly predict the series by describing the three types of data. A cubic polynomial is used to approximate each local region of a time series, and the cubic polynomial coefficients are divided into three groups, which are used to achieve high-accuracy fitting of three types of data. Fitting accuracy of the new method is comparable to cubic polynomials. Extensive experiments on six real-world datasets. demonstrate that the proposed method has remarkable potential in time series prediction.
Based on the primitive factorization theorem, this paper presents an improved algorithm for computing free bases of syzygy modules of bivariate polynomial matrices, which additionally enables efficient computation of μ -bases for rational parametric surfaces. Experimental results show that the new algorithm outperforms two existing algorithms in terms of computational efficiency. Furthermore, by leveraging this algorithm, we generalize the general matrix factorization theory of full-rank bivariate polynomial matrices to the rank-deficient case for the first time.
We investigate certain almost split sequences in the category of morphisms between projective modules over an Artin algebra. This category has interesting properties and is closely linked to τ -tilting theory and the Auslander–Reiten theory of the module category. We give an explicit description of particular almost split sequences that begin or end at specific objects, and discuss applications, including their connection to g-vectors.
In this paper, we provide some rigidity results for compact manifolds with smooth boundary by using appropriate geometric or topological assumptions and the Obata-type equation ∇ ^2 f -fg =0 with Robin boundary condition f_ν = cf , where c>1 (note that in this case f has no critical points). By the same idea, we also use equations ∇ ^2 f +fg =0 and ∇ ^2 f=0 to establish similar rigidity results. The proofs of our rigidity results are mainly based on the warped product structures on compact manifolds determined by the Obata-type equation (see Proposition 1.11), and we also provide the corresponding structure on complete non-compact manifolds with compact boundary (see Proposition 1.13). It should be pointed that we have actually provided all possible structures determined by the equation ∇ ^2 f -fg =0 with f_ν = cf for complete manifolds with compact boundary.
Bivariate interval-censored failure time data occur in many areas such as biomedical and epidemiological research and their regression analysis has been discussed by many authors. However, most of the existing methods have some limitations on either data structures or the assumed models. In this paper, we propose a sieve maximum likelihood estimation approach for general interval-censored data under a class of flexible copula-based semiparametric partly linear transformation models. The approach allows for potential nonlinear effects and makes use of Bernstein polynomials. The resulting estimators of regression parameters are shown to be consistent and asymptotically efficient and normal. A simulation study is conducted to assess the finite-sample performance of the proposed approach and suggests that it works well in practice. Also we apply it to a set of real data arising from dental research.
This paper focuses on the hydrodynamic limit problem from a kinetic Cucker–Smale model with a confining potential toward the self-organized hydrodynamic system. This paper aims to establish a uniform-in- ε convergence rate for the asymptotic expansion ansatz, and hence significantly improves the result obtained in Jiang–Luo–Zhang (Math Models Methods Appl Sci 34:2395–2467, 2024). The key ingredient of the proof is to consider a new decomposition for the remainder function, by handling the contribution of the macroscopic density separately. Moreover, the convergence holds valid in any higher-order Sobolev spaces with the general regularity index.
We developed a significance test method for multinomial naive Bayes classifier with ultra-high-dimensional binary features. A novel test statistic with asymptotic standard Gaussian null distribution is proposed. Under very mild assumptions, the proposed test statistic has powers that tend to 1 as the sample size tends to infinity. Then, a sequential test process is developed to perform variable screening. We applied the proposed methods to lots of numerical studies including simulated examples and two real text data classification examples. The results show that our methods have good finite sample performances.
In this paper, we study several types of time-restricted sensitivity both in measure-theoretical and topological settings for amenable group actions. We obtain some quantitative results between restricted asymptotic rates and entropies.
Fair curves are widely used in geometric design, modeling, and industrial manufacturing. This paper proposes a new curve fairing algorithm based on the smoothed target curvature normal. To accomplish this, the problem of constructing the target curve is converted into solving a second-order differential equation with boundary value constraints, which is then solved through the Rayleigh–Ritz method. The effectiveness of this approach is demonstrated through several illustrative examples.
In this paper, we investigate a test for parameter instability with an unknown change point in the low-rank tensor regression model. Assuming the tensor coefficient admits the CP decomposition, we embed the low-rank structure into the tensor regression, ensuring structural preservation and computational feasibility. To estimate the parameters, we construct the full-sample and partial-sample OLS estimators and prove their consistency under the null hypothesis. Based on these, we propose the quasi-likelihood ratio test statistic for detecting non-stationarity and derive its asymptotic distribution under the null hypothesis as the supremum of the square of a standardized tied-down Bessel process. Additionally, we explore the asymptotic local power under local alternatives, demonstrating that our test is a powerful tool for structural change detection. The results of simulation studies and empirical applications confirm the efficiency of the proposed method.
This paper deals with the large deviation principle of the fractional stochastic FitzHugh–Nagumo systems on ℝ^n with superlinear drift. We first establish the well-posedness and uniform tail-estimates of solutions to the controlled system associated with the original stochastic equations. We then prove the strong convergence of solutions to the controlled system with respect to the weak topology of controls by the method of tail-ends estimates in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. We finally show the large deviation principle of the FitzHugh–Nagumo system by the weak convergence method.
In large-scale multiple hypothesis testing, controlling the false discovery exceedance (FDX) is emerging as a compelling alternative to the widely employed false discovery rate (FDR) when the false discovery proportion (FDP) exhibits high variability. Existing methods for FDX control have mainly focused on nondirectional discovery within a two-group model framework. However, this approach can significantly compromise the reliability of FDX control when directional decisions are required, especially in cases when the signal-to-noise ratio is low. Furthermore, the theoretical optimality of FDX control has not been thoroughly explored. In this paper, we introduce an empirical Bayes approach tailored for directional FDX control within a three-group model. We demonstrate that an oracle decision rule, which ranks and thresholds a directional version of the local false discovery rate (lfdr), achieves optimality by maximizing power while adhering to the directional FDX constraint. We also propose a data-driven procedure to mimic the oracle rule in practical applications and establish its asymptotic optimality. Through extensive simulation studies and a real-data application, we show the superior performance of our method in directional FDX control.
The Prediction of future observations with constraints is a fundamental problem in applied statistics. In this paper, we consider incorporating parameter constraints into the frequentist, Bayesian, fiducial and inferential model (IM) prediction frameworks. As two simple examples, the constrained Gaussian and Poisson models often appear in high energy physics and we use these two models to introduce constrained prediction methods. Since the prediction interval (PI) is a useful tool for predicting future data, our simulation studies show that the PIs of fiducial and IM have better coverage performance than the frequentist and Bayesian PIs. We also discuss the use of the fiducial and IM PIs. Finally, two real examples are used to demonstrate the application of different methods.
This article proposes a new likelihood ratio-based index (LR index for short), to measure the dependence between a categorical response variable and a continuous predictor variable. The LR index is nonnegative and is zero if and only if the variables are independent. We propose an estimate of the index, develop a novel independence test and derive the asymptotic null distribution. Next, based on the LR index, a feature screening procedure (LR-SIS for short) is developed for multiclass classification with ultrahigh-dimensional predictors. LR-SIS is model-free and robust to the heavy-tailed distribution of predictors and outliers. The sure screening property of LR-SIS is established allowing the number of response classes to be diverging. The finite sample performance of the proposed LR index in both independence testing and feature screening is demonstrated by comprehensive simulation studies. Application of the LR-SIS is also illustrated on a real data set.
