Variation Regularization Research Articles

On the existence of solutions to adversarial training in multiclass classification

Abstract Adversarial training is a min-max optimization problem that is designed to construct robust classifiers against adversarial perturbations of data. We study three models of adversarial training in the multiclass agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in their previous work [21]. In addition, we develop new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we provide an alternative proof of the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting.

A hybrid overlapping group sparsity denoising model with fractional-order total variation and non-convex regularizer

A hybrid overlapping group sparsity denoising model with fractional-order total variation and non-convex regularizer

The effect of cuboid explosive shape on the dimension of damaged area in masonry walls subjected to contact explosions

Cuboid explosives are commonly available in both military and civilian markets. Masonry walls are among the most prevalent structural elements utilized in buildings worldwide. This study numerically investigates the response of masonry walls subjected to blast loads generated by cuboid explosives. The LS-DYNA software was employed to create numerical models, which were subsequently validated against experimental data. The shape of the cuboid explosives is characterized by its three-dimensional dimensions: length ([Formula: see text]), width ([Formula: see text]), and height ([Formula: see text]). To maintain a constant mass of explosives, two distinct scenarios were considered. In the first scenario, the explosive’s shape was altered by adjusting the height-to-length ratio ([Formula: see text]) while keeping [Formula: see text] equal to [Formula: see text]. In the second scenario, the length-to-width ratio ([Formula: see text]) was modified while maintaining a constant height ([Formula: see text]), leading to changes in the explosive’s shape. The effects of both the [Formula: see text] ratio and the [Formula: see text] ratio on the dynamic responses of the masonry walls were investigated. The results indicate that the dimensions of both the ejection crater and the spalling crater on the masonry wall initially increased and then decreased as the [Formula: see text] ratio increased. Furthermore, the size of the breached hole gradually diminished with an increasing [Formula: see text] ratio. In terms of changes to the [Formula: see text] ratio, the damage area of the masonry wall took on an elliptical shape, with the dimensions of this elliptical damage area exhibiting regular variations. This study aimed to equate cuboid explosives (with [Formula: see text] and [Formula: see text]) to cubic explosives (where [Formula: see text]) of the same mass. The equivalent cubic shape coefficients ([Formula: see text]) for the cuboid explosives were evaluated and calculated based on the dimensions of the damage area on the masonry wall. Furthermore, four coefficients ([Formula: see text] to [Formula: see text]) were proposed to estimate the dimensions of the masonry wall damage area when subjected to contact explosions from arbitrarily shaped cuboid explosives. The applicability of these coefficients was validated through numerical simulations.

Analysis of Mechanical Properties of Components Produced by Additive Manufacturing

This paper analyses the effect of various lattice structures and lattice vectors on the mechanical properties of components made with acrylonitrile butadiene styrene (ABS) material by fused deposition modeling (FDM) technique. ABS is a well-known thermoplastic polymer that is being used in a variety of applications because of its high durability and good adaptability. A total of nine varieties of lattice structures are examined, including regular variations like hexagonal and cubic patterns and more complex ones like Gyroid patterns in this paper. Lattice vectors like 4–6 mm were considered while preparing the components. The densities of the specimens were estimated, and the specimens were tested for their flexural strength and compression strength. The results show that “Gyroid minimal surface” structures are a better choice for flexural and compression strength applications.

Characterizing Asymptotic Dependence between a Satellite Precipitation Product and Station Data in the Northern US Rocky Mountains via the Tail Dependence Regression Framework With a Gibbs Posterior Inference Approach

ABSTRACTThe use of satellite precipitation products (SPP) allows for precipitation information to be collected nearly globally, but questions remain regarding their ability to reproduce extreme precipitation over mountainous terrain. In this work, we assess the ability of the precipitation estimation from remotely sensed information using artificial neural networks‐climate data record (PERSIANN‐CDR) to capture daily precipitation extremes by comparing PERSIANN‐CDR with corresponding station data in the summer at remote locations in the northern US Rocky Mountains of Wyoming, Idaho, and Montana. The assessment utilizes the regular variation framework from extreme value theory and consists of two parts: (1) evaluating the extent to which PERSIANN‐CDR can capture precipitation extremes through inference on an asymptotic dependence parameter, concluding that the level of asymptotic dependence is moderate throughout the region; (2) developing a tail dependence regression modeling framework and a Gibbs posterior approach for inference to investigate the degree to which elevation and topographic heterogeneity impact the level of asymptotic dependence, finding that the inclusion of a set of meteorological covariates, when combined with the PERSIANN‐CDR output, yields an increased level of asymptotic dependence with station data.

Long-range fourier ptychographic imaging of the object in multidimensional motion

Fourier Ptychographic (FP) is a cutting-edge technique for achieving high resolution in long-range imaging, holding significant research value. However, most of the research on FP has been limited to high-resolution imaging of stationary objects, considerably narrowing the scope of its applications. In real-world scenarios, the object may move in three dimensions and rotate during the image acquisition process. To address such scenarios, this paper proposes a method for achieving FP of the object in multidimensional motion using a single camera. Starting from the principles of Fourier optics and diffraction, the paper calculates the effects of an object's movement in different dimensions on the light field. The Fourier-Mellin algorithm is to be applied to deduce changes in the light field from captured intensity images and align all collected data under a chosen reference light field. During image reconstruction, we propose an additional phase retrieval algorithm that integrates total variation regularization to address aperture offset issues. The paper validates the proposed method's effectiveness through simulations and experiments. FP is successfully applied to objects in multidimensional motion. The method also doubles the imaging system's resolution.

Data-Driven Morozov Regularization of Inverse Problems

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.

Homomorphisms from Functional Equations: The Goldie Equation, II

AbstractThis first of three sequels to Homomorphisms from Functional equations: The Goldie equation (Ostaszewski in Aequationes Math 90:427–448, 2016) by the second author—the second of the resulting quartet—starts from the Goldie functional equation arising in the general regular variation of our joint paper (Bingham et al. in J Math Anal Appl 483:123610, 2020). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups—the Popa groups arising in earlier work by Popa, and their relatives the Javor groups . Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.

Flexible Krylov methods for group sparsity regularization

Abstract This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non-overlapping and overlapping groups. Group sparsity regularization refers to a type of structured sparsity regularization, where the goal is to impose additional structure in the regularization process by assigning variables to predefined groups that may represent graph or network structures. Special cases of group sparsity regularization include ℓ 1 and isotropic total variation regularization. In this work, we develop hybrid projection methods based on flexible Krylov subspaces, where we first recast the group sparsity regularization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion. Then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. The main advantages of these methods are that they are computationally efficient (leveraging the advantages of flexible methods), they are general (and therefore very easily adaptable to new regularization term choices), and they are able to select the regularization parameters automatically and adaptively (exploiting the advantages of hybrid methods). Extensions to multiple regularization terms and solution decomposition frameworks (e.g. for anomaly detection) are described, and a variety of numerical examples demonstrate both the efficiency and accuracy of the proposed approaches compared to existing solvers.

Reverse time migration of GPR data based on GPU parallel Symplectic Euler Algorithm and Total Variation Regularization

Abstract Reverse time migration is important in GPR image processing, but its efficiency and accuracy need to be improved. In terms of efficiency, reverse time migration uses the Finite Difference Time Domain (FDTD) algorithm for parallel CPU computation, which is inefficient. In terms of accuracy, it usually uses cross-correlation conditions for imaging, which reduces the imaging accuracy. To solve the above problems, this paper proposes the solution: using symplectic algorithm parallel GPU acceleration technology to improve the migration efficiency and using TV regularization technology to improve imaging accuracy. Finally, the proposed algorithm is verified by the road cavity model and the underground pipeline model.

Transitive reasoning: A high-performance computing model for significant pattern discovery in cognitive IoT sensor network

Current research on the Internet of Things (IoT) has given rise to a new field of study called cognitive IoT (CIoT), which aims to incorporate cognition into the designs of IoT systems. Consequently, the CIoT inherits specific attributes and challenges from IoT. The CIoT applications generate vast, diverse, constantly changing, and time-dependent data due to the billions of devices involved. The efficient operation of these CIoT systems requires the extraction of valuable insights from vast data sources in a computationally efficient manner. Therefore, this study proposes transitive reasoning to glean significant concepts and patterns from a 21.25-year environmental dataset. To reduce the effects of missing entries, the proposed methodology includes a grouping of data using probabilistic clustering and applying total variance regularization in the alternate direction method of multipliers (ADMM) to regularize the sensory data. As a result, noisy entries will be less conspicuous. Afterward, it calculates the transitional plausibility value for each cluster using the transited value and then turns it into binary data to create concept lattices. In addition, each concept that is formed is assigned a weight, and the concept with the largest transitive strength value is chosen, followed by calculating the mean value. Therefore, this pattern is seen as significant. Experimental results on 21.25-year environmental data show an accuracy of over 99.5%, outperforming competing methods, as shown by cross-validation using multiple metrics.

Geochemistry and Mineralogy of Phlogopite and Its Implications for Serpentinization of Jian Forsterite Jade in Southern Jilin Province, China

Mica is a kind of important rock-forming mineral in the lithosphere of Earth, which can be a superior tool used to trace the origin and late evolution of rock. The Jian forsterite jade (a kind of geological skarn) is an emerging kind of gemstone in China with a beautiful color and luster, discovered in Ji’an County, Jilin Province, Northeast China. It is mainly composed of rare Mg-rich forsterite (Mg# (Mg/(Mg + Fe2+) up to 99), serpentine and brucite. The source of hydrothermal fluid triggering the late metamorphism (the serpentinization of forsterite) of forsterite jade deposits remains unclear. We report a series of phlogopites with a regular range of mineral compositions in the forsterite jade deposit. Micrographs show that the phlogopites are associated with forsterite and coexist with serpentine in forsterite jade, tourmaline and tremolite in the contact zone, and plagioclase in pegmatite, and the related replacement of phlogopite seems to have not occurred. The phlogopites that occurred as single grains or veinlets in forsterite jade named type I are characterized by high XMg, ranging from ~0.98 to ~0.95, and the phlogopites that occurred in the contact zone of forsterite jade and pegmatite named type II are rich in Fe, with a range of XMg from ~0.82 to ~0.66. Additionally, the type II phlogopites are also rich in Ti, Mn, Cl, Li, Rb, Zn, V, Co, Nb and Ta but poor in Na, Sr and F compared to the type I phlogopite. Petrological and mineralogical characteristics and geochemical compositions suggest that the phlogopites are crystallized from the corresponding fluid component by hydrothermal metasomatism. The abundant Mg of the fluid phase is produced during the serpentinization of forsterite, triggered by pegmatitic hydrothermal fluid, and other main materials like K, Al, Si and H2O are provided by the intrusive pegmatite. With the occurrence of and regular compositional variation in phlogopites in the forsterite jade deposit, we suppose that the hydrothermal fluid triggering the serpentinization of the Jian forsterite jade is produced by the intrusive pegmatite.

Code and Data Repository for On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

The goal of this repository to be able to trace and reproduce the results and plots in the referenced article. For the results in this article, we used the general purpose integer programming solver Gurobi version 10.0.0. In addition, you need to have the python packages pandas, numpy, scipy, and pybind11 installed to be able to build the code for computing the weights of the cutting planes from 5.1.1.

On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

We analyze integer linear programs that we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regard to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling toward larger instances in the future. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant MA 10080/2-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0680 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0680 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A modified non-convex Cauchy total variation regularization model for image restoration

A modified non-convex Cauchy total variation regularization model for image restoration

Graph total variation and low-rank regularization for heterogeneous change detection

Heterogeneous change detection (HCD) is challenging because different imaging mechanisms for various sensors make images difficult to compare directly. To address this problem, a graph-based regression algorithm is proposed for HCD, by leveraging the Graph Total Variation regularization and Low-Rank matrices decomposition (GTVLR). Utilizing graph signal processing (GSP) theory, a directed graph (digraph) model is employed to effectively represent the orientation and correlation information of images, thereby enabling direct comparison of heterogeneous data within the same domain after graph filtering. The GTVLR framework facilitates the decomposition of post-event images into regression and changed images. This decomposition ensures that the regression image mirrors the structure similarity of the pre-event image, while the changed image highlights areas of alteration, aiding in change detection. The model characterizes the piecewise smoothness and Low-Rank properties of data through GTV regularization and Low-Rank penalty, respectively. Moreover, by integrating the higher-order neighboring information within the digraph to refine the model. Experiments conducted on three real-world datasets and comparison with several state-of-the-art methods demonstrate the effectiveness of the proposed algorithm.

Denoising sphere‐valued data by relaxed total variation regularization

AbstractCircle‐ and sphere‐valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore ‐sphere‐valued signals exploiting the classical anisotropic total variation on the surrounding ‐dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non‐convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0‐sphere‐valued) signals, the relaxation is provable tight, that is, the relaxed solution can be used to construct a solution of the original non‐convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)‐valued signals.

A Low Transformed Tubal Rank Tensor Model Using a Spatial-Tubal Constraint for Sample Clustering with Cancer Multi-omics Data

Background: Since each dimension of a tensor can store different types of genomics data, compared to matrix methods, utilizing tensor structure can provide a deeper understanding of multi-dimensional data while also facilitating the discovery of more useful information related to cancer. However, in reality, there are issues such as insufficient utilization of prior knowledge in multi-omics data and limitations in the recovery of low-tubal-rank tensors. Therefore, the method proposed in this article was developed. Objective: In this paper, we proposed a low transformed tubal rank tensor model (LTTRT) using a spatial-tubal constraint to accurately partition different types of cancer samples and provide reliable theoretical support for the identification, diagnosis, and treatment of cancer. Method: In the LTTRT method, the transformed tensor nuclear norm based on the transformed tensor singular value decomposition is characterized by the low-rank tensor, which can explore the global low-rank property of the tensor, resolving the challenge of the tensor nuclear norm-based method not achieving the lowest tubal rank. Additionally, the introduction of weighted total variation regularization is conducive to extracting more information from sequencing data in both spatial and tubal dimensions, exploring cross-correlation features of multiple genomic data, and addressing the problem of overlooking prior knowledge from various perspectives. In addition, the L1-norm is used to improve sparsity. A symmetric Gauss‒Seidel-based alternating direction method of multipliers (sGS-ADMM) is used to update the LTTRT model iteratively. Results: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. result: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. Conclusion: The LTTRT method achieves precise segmentation of different types of cancer samples.

Discrete wedgelet transform regularization-based spectral deconvolution for infrared spectroscopy

Infrared spectral data often exhibit band overlap and random noise when it is applied to recognize the unknown chemical materials. To address these issues, a novel regularization-based spectral deconvolution method for unknown chemical material detection (DWTSD) was proposed in this paper. The discrete wedgelet transform is introduced to analyze the difference between the latent infrared spectrum and the noisy infrared spectrum. The instrument response function is also needed to estimate simultaneously with the latent infrared spectrum. Therefore, the improved total variation regularization is introduced to constrain the smoothness of the spectral lines. Then the split Bregman iteration algorithm is also introduced to optimize the cost function. The proposed DWTSD method is simple and offers good performance with low computational load. Experimental results on simulated and real infrared spectrums show that the proposed DWTSD method has good performance in noise reduction and spectral detail generation. With the proposed methodology, the problem of instrument aging can be largely eliminated, making the reconstruction of infrared spectra a more convenient tool for the extraction of features of an unknown material and their interpretation. The applicability of the method transcends infrared spectroscopy, offering utility in a spectrum of spectroscopic analyses.

An accelerated alternating direction method of multiplier for MRI with TV regularisation

Compressed Sensing (CS) is important in the field of image processing and signal processing, and CS-Magnetic Resonance Imaging (MRI) is used to reconstruct image from undersampled k-space data. Total Variation (TV) regularisation is a common technique to improve the sparsity of image, and the Alternating Direction Multiplier Method (ADMM) plays a key role in the variational image processing problem. This paper aims to improve the quality of MRI and shorten the reconstruction time. We consider MRI to solve a linear inverse problem, we convert it into a constrained optimization problem based on TV regularisation, then an accelerated ADMM is established. Through a series of theoretical derivations, we verify that the algorithm satisfies the convergence rate of O1/k2 under the condition that one objective function is quadratically convex and the other is strongly convex. We select five undersampled templates for testing in MRI experiment and compare it with other algorithms, experimental results show that our proposed method not only improves the running speed but also gives better reconstruction results.