Proximal Operator Research Articles

Strong coupled proximal points of cyclic coupled proximal mappings using Ck-class functions in S-metric spaces

In this paper, we introduce cyclic coupled proximal mapping in S-metric spaces using Ck-class functions and prove the existence of strong coupled proximal points of such mappings in complete S-metric spaces. Also, we provide an example in support of our main result.

Information Behavior of Faculty Members of NSTU, Public University of Bangladesh

The study’s main purpose was to investigate faculty members’ information searching behaviors while administering any research. This study designed an online questionnaire and printed questionnaires used for data collection. The data were analysed using several descriptive statistics, such as frequencies, percentages, and non-parametric tests, i.e., Mann-Whitney and Kruskal-Wallis. The findings showed that faculty members were heavily dependent on search engines to access information, and they have mainly used academic social media sites such as Google Scholar (n=139) and ResearchGate (n=133). Additionally, to keep up-to-date with new publications, they primarily relied on journal alerts (n=126). In the case of applying searching strategies, they used more than one keyword search and sometimes one keyword. Conversely, they do not apply proximity operators, discovery and federated tools and Boolean operators in their search techniques. Furthermore, for modifying search techniques, they used several keywords searching and utilise search engines, databases, and advanced search techniques. Moreover, the Mann-Whitney test result found no significant differences in terms of their gender regarding the types of e-resources used by them, and the Kruskal-Wallis tests found substantial differences in terms of faculty demographic characteristics of using only indexed databases, search engines, academic, social media sites (e.g., ResearchGate, and Zotero Network), current awareness services (i.e., Journal alerts, Web alerts, and discussion lists), and search techniques (i.e., Boolean operators, and Truncation).

X-rays image reconstruction using proximal algorithm and adapted TV regularization

Computed tomography (CT) aims to reconstruct an internal distribution of an object based on projection measurements. In the case of a limited number of projections, the reconstruction problem becomes significantly ill-posed. Practically, reconstruction algorithms play a crucial role in overcoming this problem. In the case of missing or incomplete data, and in order to improve the quality of the reconstruction image, the choice of a sparse regularisation by adding l1 norm is needed. The reconstruction problem is then based on using proximal operators. We are interested in the Douglas-Rachford method and employ total variation (TV) regularization. An efficient technique based on these concepts is proposed in this study. The primary goal is to achieve high- quality reconstructed images in terms of PSNR parameter and relative error. The numerical simulation results demonstrate that the suggested technique minimizes noise and artifacts while preserving structural information. The results are encouraging and indicate the effectiveness of the proposed strategy.

Bregman primal\u2013dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

Morphometric measurement of the proximal tibia to design the tibial component of total knee arthroplasty for the Thai population

PurposeThis study evaluates the morphology of the Thai proximal tibia based on three-dimensional (3D) models to design the tibial component.MethodsThe 3D models of 480 tibias were created using reverse engineering techniques from computed tomography imaging data obtained from 240 volunteers (120 males, 120 females; range 20–50 years). Based on 3D measurements, a digital ruler was used to measure the distance between the triangular points of the models. The morphometric parameters consisted of mediolateral length (ML), anteroposterior width (AP), medial anteroposterior width (MAP), lateral anteroposterior width (LAP), central to a medial length (CM), central to a lateral length (CL), medial anterior radius (MAR), lateral anterior radius (LAR), and tibial aspect ratio (AR). An independent t-test was performed for gender differences, and K-means clustering was used to find the optimum sizes of the tibial component with a correlation between ML length and AP width in Thai people.ResultsThe average morphometric parameters of Thai proximal tibia, namely ML, AP, MAP, LAP, CM, and CL, were as follows: 72.52 ± 5.94 mm, 46.36 ± 3.84 mm, 49.22 ± 3.62 mm, 43.59 ± 4.05 mm, 14.29 ± 2.72 mm, and 15.28 ± 2.99 mm, respectively. The average of MAR, LAR, and AR was 24.43 ± 2.11 mm, 21.52 ± 2.00 mm, and 1.57 ± 0.08, respectively. All morphometric parameters in males were significantly higher than those of females. There was a difference between the Thai proximal tibia and other nationalities and a mismatch between the size of the commercial tibial component and the Thai knee. Using K-means clustering analysis, the recommended number of ML and AP is seven sizes for the practical design of tibial components to cover the Thai anatomy.ConclusionThe design of the tibial component should be recommended to cover the anatomy of the Thai population. These data provide essential information for the specific design of Thai knee prostheses.

Computing sparse cones with bounded distortion for conformal parameterizations

We propose a novel method to generate sparse cone singularities with bounded distortion constraints for conformal parameterizations. It is formulated as minimizing the ℓ 0 -norm of Gaussian curvature of vertices with hard constraints of bounding the distortion that is measured by the ℓ 2 -norm of the log conformal factor. We use the reweighted ℓ 1 -norm to approximate the ℓ 0 -norm and solve each convex weighted ℓ 1 minimization subproblem by the Douglas-Rachford (DR) splitting scheme. To quickly generate sparse cones, we modify DR splitting by weighting the ℓ 2 -norm of the proximal mapping to force the small Gaussian curvature to quickly approach zero. Accordingly, compared with the conventional DR splitting, the modified method performs one to two orders of magnitude faster. Besides, we perform variable substitution of log conformal factors to simplify the computation process for acceleration. Our algorithm is able to bound distortion to compute sparse cone singularities, so that the resulting conformal parameterizations achieve a favorable tradeoff between the area distortion and the number of cones. We demonstrate its effectiveness and feasibility on a large number of models.

Efficient Proximal Mapping Computation for Low-Rank Inducing Norms

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.

L1-L2 norm regularization via forward-backward splitting for fluorescence molecular tomography.

Fluorescent molecular tomography (FMT) is a highly sensitive and noninvasive imaging approach for providing three-dimensional distribution of fluorescent marker probes. However, owing to its light scattering effect and the ill-posedness of inverse problems, it is challenging to develop an efficient reconstruction algorithm that can achieve the exact location and morphology of the fluorescence source. In this study, therefore, in order to satisfy the need for early tumor detection and improve the sparsity of solution, we proposed a novel L 1-L 2 norm regularization via the forward-backward splitting method for enhancing the FMT reconstruction accuracy and the robustness. By fully considering the highly coherent nature of the system matrix of FMT, it operates by splitting the objective to be minimized into simpler functions, which are dealt with individually to obtain a sparser solution. An analytic solution of L 1-L 2 norm proximal operators and a forward-backward splitting algorithm were employed to efficiently solve the nonconvex L 1-L 2 norm minimization problem. Numerical simulations and an in-vivo glioma mouse model experiment were conducted to evaluate the performance of our algorithm. The comparative results of these experiments demonstrated that the proposed algorithm obtained superior reconstruction performance in terms of spatial location, dual-source resolution, and in-vivo practicability. It was believed that this study would promote the preclinical and clinical applications of FMT in early tumor detection.

Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

Due to the importance of Yosida approximation operator, we generalized the variational inequality problem and its equivalent problems by using Yosida approximation operator. The aim of this work is to introduce and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. We prove an existence result together with convergence analysis for Yosida proximal operator equation involving XOR-operation. For this purpose, we establish an algorithm based on fixed point formulation. Our approach is based on a proximal operator technique involving a subdifferential operator. As an application of our main result, we provide a numerical example using the MATLAB program R2018a. Comparing different iterations, a computational table is assembled and some graphs are plotted to show the convergence of iterative sequences for different initial values.

Inexact accelerated high-order proximal-point methods

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate Oleft( k^{-4}right) , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence Oleft( k^{-(3p+1)/ 2}right) , where p ge 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

A First-Order Optimization Algorithm for Statistical Learning with Hierarchical Sparsity Structure

In many statistical learning problems, it is desired that the optimal solution conform to an a priori known sparsity structure represented by a directed acyclic graph. Inducing such structures by means of convex regularizers requires nonsmooth penalty functions that exploit group overlapping. Our study focuses on evaluating the proximal operator of the latent overlapping group lasso developed by Jacob et al. in 2009 . We implemented an alternating direction method of multiplier with a sharing scheme to solve large-scale instances of the underlying optimization problem efficiently. In the absence of strong convexity, global linear convergence of the algorithm is established using the error bound theory. More specifically, the paper contributes to establishing primal and dual error bounds when the nonsmooth component in the objective function does not have a polyhedral epigraph. We also investigate the effect of the graph structure on the speed of convergence of the algorithm. Detailed numerical simulation studies over different graph structures supporting the proposed algorithm and two applications in learning are provided. Summary of Contribution: The paper proposes a computationally efficient optimization algorithm to evaluate the proximal operator of a nonsmooth hierarchical sparsity-inducing regularizer and establishes its convergence properties. The computationally intensive subproblem of the proposed algorithm can be fully parallelized, which allows solving large-scale instances of the underlying problem. Comprehensive numerical simulation studies benchmarking the proposed algorithm against five other methods on the speed of convergence to optimality are provided. Furthermore, performance of the algorithm is demonstrated on two statistical learning applications related to topic modeling and breast cancer classification. The code along with the simulation studies and benchmarks are available on the corresponding author’s GitHub website for evaluation and future use.

Finding best approximation pairs for two intersections of closed convex sets

The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history that dates back to work by Cheney and Goldstein in 1959. In 2018, Aharoni, Censor, and Jiang revisited this problem and proposed an algorithm that can be used when the two sets are finite intersections of halfspaces. Motivated by their work, we present alternative algorithms that utilize projection and proximity operators. Our modeling framework is able to accommodate even convex sets. Numerical experiments indicate that these methods are competitive and sometimes superior to the one proposed by Aharoni et al.

L1-L2 Minimization Via A Proximal Operator For Fluorescence Molecular Tomography.

Fluorescent Molecular Tomography (FMT) is a highly sensitive and noninvasive imaging method that provides three-dimensional distribution of biomarkers by noninvasive detection of fluorescent marker probes. However, due to the light scattering effect and ill-posedness of inverse problems, it is challenging to develop an efficient construction method that can provide the exact location and morphology of the fluorescence distribution. In this paper, we proposed L1-L2 norm regularization to improve FMT reconstruction. In our research, proximal operators of non-convex L1 -L2 norm and forward-backward splitting method was adopted to solve the inverse problem of FMT. Simulation results on heterogeneous mouse model demonstrated that the proposed FBS method is superior to IVTCG, DCA and IRW-L1/2 reconstruction methods in location accuracy and other aspects.

Ptychography-based high-throughput lensless on-chip microscopy via incremental proximal algorithms.

Ptychography-based lensless on-chip microscopy enables high-throughput imaging by retrieving the missing phase information from intensity measurements. Numerous reconstruction algorithms for ptychography have been proposed, yet only a few incremental algorithms can be extended to lensless on-chip microscopy because of large-scale datasets but limited computational efficiency. In this paper, we propose the use of accelerated proximal gradient methods for blind ptychographic phase retrieval in lensless on-chip microscopy. Incremental gradient approaches are adopted in the reconstruction routine. Our algorithms divide the phase retrieval problem into sub-problems involving the evaluation of proximal operator, stochastic gradient descent, and Wirtinger derivatives. We benchmark the performances of accelerated proximal gradient, extended ptychographic iterative engine, and alternating direction method of multipliers, and discuss their convergence and accuracy in both noisy and noiseless cases. We also validate our algorithms using experimental datasets, where full field of view measurements are captured to recover the high-resolution complex samples. Among these algorithms, accelerated proximal gradient presents the overall best performance regarding accuracy and convergence rate. The proposed methods may find applications in ptychographic reconstruction, especially for cases where a wide field of view and high resolution are desired at the same time.

A proximal LAVA method for genome-wide association and prediction of traits with mixed inheritance patterns

BackgroundThe genetic basis of phenotypic traits is highly variable and usually divided into mono-, oligo- and polygenic inheritance classes. Relatively few traits are known to be monogenic or oligogeneic. The majority of traits are considered to have a polygenic background. To what extent there are mixtures between these classes is unknown. The rapid advancement of genomic techniques makes it possible to directly map large amounts of genomic markers (GWAS) and predict unknown phenotypes (GWP). Most of the multi-marker methods for GWAS and GWP falls into one of two regularization frameworks. The first framework is based on ell _1-norm regularization (e.g. the LASSO) and is suitable for mono- and oligogenic traits, whereas the second framework regularize with the ell _2-norm (e.g. ridge regression; RR) and thereby is favourable for polygenic traits. A general framework for mixed inheritance is lacking.ResultsWe have developed a proximal operator algorithm based on the recent LAVA regularization method that jointly performs ell _1- and ell _2-norm regularization. The algorithm is built on the alternating direction method of multipliers and proximal translation mapping (LAVA ADMM). When evaluated on the simulated QTLMAS2010 data, it is shown that the LAVA ADMM together with Bayesian optimization of the regularization parameters provides an efficient approach with lower test prediction mean-squared-error (65.89) than the LASSO (66.11), Ridge regression (83.41) and Elastic net (66.11). For the real pig data the test MSE of the LAVA ADMM is 0.850 compared to the LASSO, RR and EN with 0.875, 0.853 and 0.853, respectively.ConclusionsThis study presents the LAVA ADMM that is capable of joint modelling of monogenic major genetic effects and polygenic minor genetic effects which can be used for both genome-wide assoiciation and prediction purposes. The statistical evaluations based on both simulated and real pig data set shows that the LAVA ADMM has better prediction properies than the LASSO, RR and EN. Julia code for the LAVA ADMM is available at: https://github.com/patwa67/LAVAADMM.

SDDRO-PBS: Simultaneous Dose and Dose Rate Optimization for FLASH Proton SBRT Based on Pencil Beam Scanning Dose Rate Modeling

Although the FLASH therapy (FLASH-RT) can reduce radiation-induced normal tissue toxicities while maintaining tumor response, its effectiveness for general cancer patients is to be further validated through clinical trials. The proton therapy is a natural choice for clinical FLASH-RT, as isochronous cyclotron-based proton systems are the only commercially available devices that can deliver ultra-high dose rates (e.g., 40 Gy/s or more) required for general-purpose clinical FLASH-RT. However, the state-of-the-art treatment planning method via intensity modulated proton therapy (IMPT) only optimizes the dose distribution and does not directly optimize the dose rate. This work aims to develop a new treatment planning method with FLASH-dose-rate optimization capability that is compatible with clinical FLASH-RT.The dose rate modeling in this work is based on pencil beam scanning (PBS) dose rate, which accounts for the spatiotemporal delivery patterns of PBS and the beam-off time between multiple spots. The planning objectives for SDDRO-PBS include both dose-rate-volume and dose-volume constraints. Various machine constraints are enforced during plan optimization for plan deliverability and efficiently solved via proximal operators, including maximum beam intensity constraint, minimum constraint for spot weights per energy layer, overall machine minimum constraint for spot weight thresholds. Setup and range uncertainties are modelled by probabilistic formulation and solved by robust optimization. Mathematically, SDDRO-PBS solves an inverse optimization problem for deliverable PBS spot weights and per-energy-layer minimum-spot thresholds. Effective optimization algorithms are developed to deal with the nonconvexity and nonlinearity of SDDRO-PBS, using iterative convex relaxations powered by alternating direction method of multipliers.Three hypofractionated or SBRT lung cases were considered to validate SDDRO-PBS in comparison with IMPT. For fair comparison, all six plans under comparison were optimized with the same dose-volume objectives, optimization algorithms, and plan normalization. Owning to its dose rate optimization capability, SDDRO-PBS had additional dose-rate-volume objectives and aimed to achieve FLASH dose rate in the high-dose regions of interest (ROI) surrounding tumor targets. The results suggest that SDDRO-PBS had comparable dose distribution and substantially improved FLASH dose rate coverage from IMPT. In particular, SDDRO-PBS achieved 95% or higher FLASH-dose-rate coverage in high-dose ROI. Moreover, the increase of dose per fraction further improved the plan quality in terms of both dose and dose rate distributions.A new and compatible treatment planning method SDDRO-PBS for clinical FLASH-RT is developed with FLASH-dose-rate optimization capability.H. Gao: None. M.M. Folkerts: None. Y. Lin: None. E. Abel: None. J.D. Bradley: Honoraria; Genentech, Inc, Mevion Medical Systems. Consultant; AstraZeneca, Inc, Varian Medical Systems, Inc. Advisory Board; Genentech, Inc, Mevion Medical Systems; American Radium Society Executive Committee. Organize NRG Oncology research agenda on lung cancer; American College of Radiology.

On Seven Fundamental Optimization Challenges in Machine Learning

Many recent successes of machine learning went hand in hand with advances in optimization. The exchange of ideas between these fields has worked both ways, with machine learning building on standard optimization procedures such as gradient descent, as well as with new directions in the optimization theory stemming from machine learning applications. In this thesis, we discuss new developments in optimization inspired by the needs and practice of machine learning, federated learning, and data science. In particular, we consider seven key challenges of mathematical optimization and develop a solution to each. Our first contribution is the resolution of a key open problem in Federated Learning: we establish the first theoretical guarantees for the famous Local SGD algorithm in the heterogeneous data regime. As the second challenge, we close the gap between the upper and lower bounds for the theory of two algorithms known as Random Reshuffling (RR) and Shuffle-Once that are widely used in practice, and set as the default data selection strategies for SGD in modern machine learning software. Our third contribution can be seen as a combination of our new theory for proximal RR and Local SGD yielding a new algorithm, which we call FedRR. Unlike Local SGD, FedRR can provably beat gradient descent in communication complexity in the heterogeneous data regime. The fourth challenge is related to the class of adaptive methods. In particular, we present the first parameter-free stepsize rule for gradient descent that provably works for any locally smooth convex objective. The fifth challenge is the development of an algorithm for distributed optimization with quantized updates that preserves linear convergence of gradient descent. Finally, in our sixth and seventh challenges, we develop new VR mechanisms applicable to the non-smooth setting based on proximal operators.

Consensus Equilibrium for Subsurface Delineation

Heterogeneity and insufficient site characterization limit our knowledge of the subsurface. Inversion techniques, which minimize the mismatch between observations and model predictions, have become an essential tool of subsurface characterization. Most optimization-based approaches fail to incorporate various implicit priors and capture the geological complexity. We overcome these limitations by deploying the plug-and-play and consensus equilibrium (CE) strategies, which provide a flexible framework for image reconstruction. Our CE methodology for spatial delineation of geologic formations consists of an image denoiser and a variational auto-encoder (deep learning-based emulator). The former ameliorates the reconstruction noise, yielding well-defined geological structures; its mathematical equivalence with the proximal operator allows the deployment of advanced denoisers (e.g., CNN-based denoiser) that do not correspond to a regularization objective. The latter defines a geology prior that imposes a geological constraint, for example, continuity and shape of geological features, onto the reconstructed image. We conduct a series of numerical experiments dealing with transient two-dimensional flow driven by a pumping well and natural hydraulic head gradient. They demonstrate the CE framework's ability to delineate, both probabilistically and deterministically, complex subsurface environments with sufficient quality.

Polaris-LAMP: Multi-Modal 3-D Image Reconstruction With a Commercial Gamma-Ray Imager

The Polaris-LAMP multi-modal 3-D gamma-ray imager is a radiation mapping and imaging platform which uses a commercial off-the-shelf (COTS) detector integrated with a contextual sensor localization and mapping platform. The integration of these systems enables a free-moving radiation imaging capability with proximity mapping, coded-aperture, and Compton imaging modalities, which can create 3-D reconstruction of photon sources from tens of keV to several MeV. Gamma-ray events are recorded using a segmented cadmium zinc telluride (CZT) detector (Polaris-H Quad by H3D Inc., Ann Arbor, MI, USA), while scene data are derived from a contextual sensor and computation package developed by Lawrence Berkeley National Laboratory which includes GPS, laser ranging, and inertial measurement sensors. An onboard computer uses these inputs to create rapidly updating pose (10 Hz) and 3-D scene estimates using a simultaneous localization and mapping (SLAM) algorithm. The precise gamma-ray event location and timing resolution of the Polaris CZT sensor enables Compton imaging above several hundred keV, while photon sources at lower images are localized using coded-aperture imaging techniques. The multi-modal imaging concept enables imaging of diverse radiation sources spanning from the 59-keV emission of <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">241</sup> Am to the 1.1 and 1.3 MeV lines of <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">60</sup> Co. This work focuses on the description of the operational principles of the detector system and demonstrating the 3-D imaging performance in a variety of source detection and mapping scenarios. As a proof of concept, we demonstrate mapping complex environments, including both point source and distributed-source environments using proximity, coded-aperture, and Compton imaging modalities. Furthermore, we show the successful use of the system to perform measurements in high-background environments through analysis of arrays of uranium hexafluoride cylinders at the Paducah UF6 project site.

GWAS Meta-Analysis Reveals Shared Genes and Biological Pathways between Major Depressive Disorder and Insomnia.

Major depressive disorder (MDD) is one of the most prevalent and disabling mental disorders worldwide. Among the symptoms of MDD, sleep disturbance such as insomnia is prominent, and the first reason patients may seek professional help. However, the underlying pathophysiology of this comorbidity is still elusive. Recently, genome-wide association studies (GWAS) have begun to unveil the genetic background of several psychiatric disorders, including MDD and insomnia. Identifying the shared genomic risk loci between comorbid psychiatric disorders could be a valuable strategy to understanding their comorbidity. This study seeks to identify the shared genes and biological pathways between MDD and insomnia based on their shared genetic variants. First, we performed a meta-analysis based on the GWAS summary statistics of MDD and insomnia obtained from Psychiatric Genomics Consortium and UK Biobank, respectively. Next, we associated shared genetic variants to genes using two gene mapping strategies: (a) positional mapping based on genomic proximity and (b) expression quantitative trait loci (eQTL) mapping based on gene expression linkage across multiple tissues. As a result, a total of 719 shared genes were identified. Over half (51%) of them are protein-coding genes. Functional enrichment analysis shows that the most enriched biological pathways are related to epigenetic modification, sensory perception, and immunologic signatures. We also identified druggable targets using a network approach. Together, these results may provide insights into understanding the genetic predisposition and underlying biological pathways of comorbid MDD and insomnia symptoms.