Variation Norm Research Articles

Synthesis of Contiguously Subarrayed Arrays With Geometry Control via Alternating Total Variation Norm Optimization

Synthesis of Contiguously Subarrayed Arrays With Geometry Control via Alternating Total Variation Norm Optimization

HIGHLY ADAPTIVE LASSO: MACHINE LEARNING THAT PROVIDES VALID NONPARAMETRIC INFERENCE IN REALISTIC MODELS.

Understanding treatment effects on health-related outcomes using real-world data requires defining a causal parameter and imposing relevant identification assumptions to translate it into a statistical estimand. Semiparametric methods, like the targeted maximum likelihood estimator (TMLE), have been developed to construct asymptotically linear estimators of these parameters. To further establish the asymptotic efficiency of these estimators, two conditions must be met: 1) the relevant components of the data likelihood must fall within a Donsker class, and 2) the estimates of nuisance parameters must converge to their true values at a rate faster than . The Highly Adaptive LASSO (HAL) satisfies these criteria by acting as an empirical risk minimizer within a class of càdlàg functions with a bounded sectional variation norm, which is known to be Donsker. HAL achieves the desired rate of convergence, thereby guaranteeing the estimators' asymptotic efficiency. The function class over which HAL minimizes its risk is flexible enough to capture realistic functions while maintaining the conditions for establishing efficiency. Additionally, HAL enables robust inference for non-pathwise differentiable parameters, such as the conditional average treatment effect (CATE) and causal dose-response curve, which are important in precision health. While these parameters are often considered in machine learning literature, these applications typically lack proper statistical inference. HAL addresses this gap by providing reliable statistical uncertainty quantification that is essential for informed decision-making in health research.

Robust support function machines for set-valued data classification

Support function machines (SFMs) have been proposed to handle set-valued data, but they are sensitive to outliers and unstable for re-sampling due to the use of the hinge loss function. To address these problems, we propose a robust SFM model with proximity functions. We first define a family of proximity functions that are used to convert set-valued data into continuous functions in a Banach space, and then we use the margin maximization in a Banach space to construct the pinball SFMs (PinSFMs). We study some properties of the proposed model, and it is interesting to observe that the optimal measure of the proposed model has a specific representation under the total variation norm. Using the representation of the optimal measure, we convert an infinite-dimensional optimization problem into a finite-dimensional optimization problem. Unlike SFMs, we employ a sampling strategy to tackle the finite-dimensional optimization problem. We theoretically show that the sparse solution determines the sparsity of the sampling points though the sampling strategy causes uncertainty for the sampling points. In addition, we achieve kernel versions of proximity functions, and the attractive property of this kernelization is that the proposed model is convex even if indefinite kernels are employed. Experiments on a series of data sets are performed to demonstrate that the proposed model is superior to some existing models in the presence of outliers.

Joint local smoothness and low-rank tensor representation for robust multi-view clustering

Low-rank tensor representation (LRTR) has become a significant method for achieving improved multi-view clustering (MVC) performance. Generally, most LRTR methods impose a tensor low-rank constraint (TLRC) on a tensor, which is spliced by the representation matrix of each view, to explore the low-rank prior hidden in these representation matrices. However, two problems remain unsolved. On the one hand, these representation matrices still contain noise since the raw data usually possess noise; on the other hand, and more importantly, is there any other prior information that can be explored among these representation matrices? Since samples located in the same cluster are similar, the coefficients of their representation matrices should exhibit similar. That is, these representation matrices should be local smoothness (LS), which is also verified by our numerical tests. In addition, the use of the LS prior helps to denoise the noise contain in the data. Then, in this paper, we propose a joint LS and LRTR for robust MVC method (LS-LRTR), which can simultaneously exploit the low-rank and LS priors. Specifically, we utilize a TLRC to explore the low-rank prior in the representation matrices. Subsequently, to mine the LS prior and further reduce the influence of noise, we introduce the Total Variation (TV) norm to the constraint representation matrices. Then, we fuse the TLRC and TV norm into a unified framework. Additionally, we apply an Augmented Lagrange Multiplier to solve the optimization problem of LS-LRTR. Experiments conducted on several datasets indicate that LS-LRTR outperforms the state-of-the-art clustering methods.

Non-Blind Image Deblurring via Shear Total Variation Norm

In this paper, we propose a novel shear gradient operator by combining the shear and gradient operators. The shear gradient operator performs well to capture diverse directional information in the image gradient domain. Based on the shear gradient operator, we extend the total variation (TV) norm to the shear total variation (STV) norm by adding two shear gradient terms. Subsequently, we introduce a shear total variation deblurring model. Experimental results are provided to validate the ability of the STV norm to capture the detailed information. Leveraging the Block Circulant with Circulant Blocks (BCCB) structure of the shear gradient matrices, the alternating direction method of multipliers (ADMM) algorithm can be used to solve the proposed model efficiently. Numerous experiments are presented to verify the performance of our algorithm for non-blind image deblurring.

Multi-scale Sparse Domination

We prove a bilinear form sparse domination theorem that applies to many multi-scale operators beyond Calderón–Zygmund theory, and also establish necessary conditions. Among the applications, we cover large classes of Fourier multipliers, maximal functions, square functions and variation norm operators.

Deep Residual Network and Wavelet Transform-Based Non-Local Means Filter for Denoising Low-Dose Computed Tomography

Image denoising helps to strengthen the image statistics and the image processing scenario. Because of the inherent physical difficulties of various recording technologies, images are prone to the emergence of some noise during image acquisition. In the existing methods, poor illumination and atmospheric conditions affect the overall performance. To solve these issues, in this paper Political Taylor-Anti Coronavirus Optimization (Political Taylor-ACVO) algorithm is developed by integrating the features of Political Optimizer (PO) with Taylor series and Anti Coronavirus Optimization (ACVO). The input medical image is subjected to noisy pixel identification step, in which the deep residual network (DRN) is used to discover noise values and then pixel restoration process is performed by the created Political Taylor-ACVO algorithm. Thereafter image enhancement mechanism strategy is done using vectorial total variation (VTV) norm. On the other hand, original image is applied to discrete wavelet transform (DWT) such that transformed result is fed to non-local means (NLM) filter. An inverse discrete wavelet transform (IDWT) is utilized to the filtered outcome for generating the denoised image. Finally, image enhancement result is fused with denoised image computed through filtering model to compute fused output image. The proposed model observed the value for Peak signal-to-noise ratio (PSNR) of 29.167 dB, Second Derivative like Measure of Enhancement (SDME) of 41.02 dB, and Structural Similarity Index (SSIM) of 0.880 for Gaussian noise.

Optimal Rates of Approximation by Shallow ReLU$$^k$$ Neural Networks and Applications to Nonparametric Regression

AbstractWe study the approximation capacity of some variation spaces corresponding to shallow ReLU$$^k$$ k neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation rates in terms of the number of neurons for shallow ReLU$$^k$$ k neural networks. It is also shown how these results can be used to derive approximation bounds for deep neural networks and convolutional neural networks (CNNs). As applications, we study convergence rates for nonparametric regression using three ReLU neural network models: shallow neural network, over-parameterized neural network, and CNN. In particular, we show that shallow neural networks can achieve the minimax optimal rates for learning Hölder functions, which complements recent results for deep neural networks. It is also proven that over-parameterized (deep or shallow) neural networks can achieve nearly optimal rates for nonparametric regression.

Extreme Few-View Tomography without Training Data.

There are fewer than 10 projection views in extreme few-view tomography. The state-of-the-art methods to reconstruct images with few-view data are compressed sensing based. Compressed sensing relies on a sparsification transformation and total variation (TV) norm minimization. However, for the extreme few-view tomography, the compressed sensing methods are not powerful enough. This paper seeks additional information as extra constraints so that extreme few-view tomography becomes possible. In transmission tomography, we roughly know the linear attenuation coefficients of the objects to be imaged. We can use these values as extra constraints. Computer simulations show that these extra constraints are helpful and improve the reconstruction quality.

Stability and dynamical bifurcation of a stochastic regime-switching predator–prey model

This paper considers a stochastic regime-switching predator–prey system with general functional response. We show that the system exactly possesses two dynamical bifurcation points (DBPs), and the expressions of these DBPs are given explicitly. In addition, we provide the sharp sufficient criteria under which the system admits a unique ergodic invariant measure (UEIM), and show that the transition function of the solution of the system exponentially converges to the UEIM under the total variation norm. The findings show that the stability and dynamical bifurcation of the system have close relationships with the random perturbations. Additionally, the main findings are applied to several special cases, and a number of recent reports are improved.

Double stranded smoothing technique in energy minimization problems

<p style='text-indent:20px;'>In this study, we propose a new algorithm for image denoising by using double stranded Bezier curves based smoothing technique. The algorithm is developed to solve energy minimization problems with non-smooth total variation norm (TV). In order to demonstrate the efficiency of the algorithm we apply it to some numerical examples. The experimental results are reported and they are compared with the results obtained from TV algorithm with hyperbolic smoothing and standard TV algorithm.</p>

L0 Gradient-Regularization and Scale Space Representation Model for Cartoon and Texture Decomposition.

In this paper, we consider decomposing an image into its cartoon and texture components. Traditional methods, which mainly rely on the gradient amplitude of images to distinguish between these components, often show limitations in decomposing small-scale, high-contrast texture patterns and large-scale, low-contrast structural components. Specifically, these methods tend to decompose the former to the cartoon image and the latter to the texture image, neglecting the scale features inherent in both components. To overcome these challenges, we introduce a new variational model which incorporates an L0-based total variation norm for the cartoon component and an L2 norm for the scale space representation of the texture component. We show that the texture component has a small L2 norm in the scale space representation. We apply a quadratic penalty function to handle the non-separable L0 norm minimization problem. Numerical experiments are given to illustrate the efficiency and effectiveness of our approach.

Iterative Image Reconstruction with Under-Sampled Data Assisted by a Neural Network.

Image reconstruction with under-sampled data is usually achieved by an iterative algorithm, which minimizes an objective function. The objective function commonly contains a data fidelity term and one or more Bayesian terms. A popular Bayesian term is the total variation (TV) norm of the image. This paper suggests an addition Bayesian term that is generated by a neural network. This neural network is essentially a classifier. This classifier can recognize the artifacts caused by the incomplete data. This classifier is trained by patient images reconstructed by complete and incomplete data sets. This newly introduced Bayesian term is referred to as the CNN score, which is a real number in (-∞, ∞). Patient studies show the good correlation between the CNN score and the severeness of the artifacts due to the incomplete measurements. A neural network can extract features from images that are suffering from incomplete measurements and convert the features into a CNN score. An iterative image reconstruction algorithm can be developed to minimize this CNN score to suppress the artifacts in the reconstructed image.

Language variation and temporary norm development in intercultural interactions

Language variation and temporary norm development in intercultural interactions

Low-tubal-rank tensor completion via local and nonlocal knowledge

Many previous studies have indicated that exploiting prior information about the underlying tensor data is a sound approach for completing missing or damaged entries. This prior information can be divided into three commonly used categories: low-rankness, local piecewise smoothness, and nonlocal self-similarity (NSS) priors. Although existing methods based on these priors have gained considerable attention, the majority of studies utilize only one or two of these priors, leading to inadequate extraction of structural information from tensors. To address this limitation and comprehensively depict the inherent structural information of underlying tensor data, this article develops a novel tensor completion framework that can simultaneously utilize the three abovementioned priors within a plug-and-play framework. More precisely, we adopt the tensor correlated total variation (t-CTV) norm as a robust representation for capturing the combined effects of low-rankness and local piecewise smoothness priors, eliminating the need for a trade-off parameter in the process; meanwhile, we introduce an advanced denoiser to explore the NSS prior. Furthermore, to address the presented optimization model, we design an alternating direction method of multipliers (ADMM) algorithm and innovatively provide its corresponding global convergence guarantees. Extensive numerical experiments on real tensor data, including color, medical and hyperspectral images, demonstrate that the proposed method surpasses various advanced approaches in terms of both quality metrics and visual effects.

One-bit SAR imaging algorithm based on MC function and TV norm

Synthetic Aperture Radar (SAR) imaging is generally characterized by a large amount of data and a high sampling rate. The traditional one-bit compressed sensing will generate virtual targets when recovering images at a low SNR, resulting in low reconstruction accuracy and obvious noise impact of the algorithm. In this paper, a one-bit SAR imaging algorithm based on the Minimax Concave (MC) penalty function and the Total Variation (TV) norm is proposed, which can potentially improve the reconstruction accuracy. At the same time, to solve the problem of speckle noise generated in the imaging process, this paper proposes a Fast Low-rank Sparse Decomposition (FLRSD) algorithm based on the one-bit contaminated image, which potentially improves the anti-noise performance and reconstruction efficiency. The results of simulation and measured data show that the proposed algorithms have better reconstruction accuracy and focusing performance than other algorithms. Even at low SNR, they also have good reconstruction effect.

Infrared Dim Small Target Detection Based on Nonconvex Constraint with L1–L2 Norm and Total Variation

Infrared dim small target detection has received a lot of attention, because it is a crucial component of the IR search and track systems (IRST). The robust principal component analysis (RPCA) is a common detection framework, which works with poor performance with complex background edges and sparse clutters due to the inappropriate approximation of sparse items. A nonconvex constraint detection method based on the difference between the L1 and L2 (L1–L2) norm and total variation (TV) is presented. The L1–L2 norm is a more accurate sparse item approximation of L0 norm, which can achieve a better description of the sparse item to separate the target from the complex backgrounds. Then, the total variation norm is conducted on the target image to suppress the sparse clutters. The new model is solved using the alternating direction method of multipliers (ADMM) method. Then, the subproblems in the model are tackled by the difference of convex algorithm (DCA) and the Newton conjugate gradient (Newton-CG) solving L1–L2 norm and TV norm, respectively. In the experiment, we conducted experiments on multiple and single target datasets, and the proposed model outperforms the state-of-the-art (SOTA) methods in terms of background suppression and robustness to accurately detect the target. It can achieve a higher true position rate (TPR) with a low false position rate (FPR).

Exponential ergodicity for singular reflecting McKean–Vlasov SDEs

By refining a recent result of Xie and Zhang (2020), we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean–Vlasov SDEs with or without reflection. The exponential ergodicity in the relative entropy and (weighted) Wasserstein distances are also studied for reflecting McKean–Vlasov SDEs. The main results are illustrated by non-symmetric singular granular media equations.

Sigma Delta Quantization for Images

AbstractIn signal quantization, it is well‐known that introducing adaptivity to quantization schemes can improve their stability and accuracy in quantizing bandlimited signals. However, adaptive quantization has only been designed for one‐dimensional signals. The contribution of this paper is two‐fold: (i) we propose the first family of two‐dimensional adaptive quantization schemes that maintain the same mathematical and practical merits as their one‐dimensional counterparts, and (ii) we show that both the traditional 1‐dimensional and the new 2‐dimensional quantization schemes can effectively quantize signals with jump discontinuities, which immediately enable the usage of adaptive quantization on images. Under mild conditions, we show that by using adaptivity, the proposed method is able to reduce the quantization error of images from the presently best to the much smaller , where s is the number of jump discontinuities in the image and P () is the total number of samples. This ‐fold error reduction is achieved via applying a total variation norm regularized decoder, whose formulation is inspired by the mathematical super‐resolution theory in the field of compressed sensing. Compared to the super‐resolution setting, our error reduction is achieved without requiring adjacent spikes/discontinuities to be well‐separated, which ensures its broad scope of application.We numerically demonstrate the efficacy of the new scheme on medical and natural images. We observe that for images with small pixel intensity values, the new method can significantly increase image quality over the state‐of‐the‐art method. © 2022 Wiley Periodicals, Inc.

Optimal Deep CNN-Based Vectorial Variation Filter for Medical Image Denoising.

Medical imaging has acquired more attention due to the emerging design of wireless technologies, the internet, and data storage. The reflection of these technologies has gained attraction in medicine and medical sciences facilitating the diagnosis and treatment of different diseases in an effective manner. However, medical images are vulnerable to noise, which can make the image unclear and perplex the identification. Thus, denoising of medical images is imperative for processing medical images. This paper devises a novel optimal deep convolution neural network-based vectorial variation (ODVV) filter for denoising medical computed tomography (CT) images and Lena images. Here, the input medical images are fed to a noisy pixel map identification module wherein the deep convolutional neural network (Deep CNN) is adapted for discovering noisy pixel maps. Here, Deep CNN training is done with the Adam algorithm. Once noisy pixels are identified, it is further given to noise removal module which is performed using the proposed optimization algorithm, namely Feedback Artificial Lion (FAL). Here, the FAL is devised by combining the FAT and Lion algorithm. After noise removal, the pixel enhancement is performed using the vectorial total variation norm to get final pixel-enhanced image. The proposed FAL algorithm offered enhanced performance in contrast to other techniques with the highest peak signal-to-noise ratio (PSNR) of 24.149dB, highest second-derivative-like measure of enhancement (SDME) of 32.142dB, highest structural index similarity (SSIM) of 0.800, and Edge Preserve Index (EPI) of 0.9267.