Expensive optimization problems pose significant challenges to traditional gradient-free optimization due to their costly evaluation overhead. Surrogate model-assisted evolutionary optimization, which substitutes expensive evaluation functions with surrogate models, can effectively overcome these challenges. Designing an efficient surrogate model is the key issue in model-assisted evolutionary optimization. In recent years, establishing surrogate models through the relationships between solutions has become a promising modeling strategy, following regression and classification models. However, there has been a notable lack of systematic organization or comprehensive summary of relation models, which has impeded the structured development of this burgeoning research area. This paper seeks to address this gap by viewing relations as a perspective to outline the contextual development of the field, defining a robust framework for researching relation models, and reviewing typical strategies within each framework. Finally, it validates the effectiveness of numerous strategies through experiments. The entire collection of strategies will be open-sourced on GitHub1, facilitating greater participation from the research community in this field of study.

Tensors play a crucial role in numerous scientific and engineering fields. This paper addresses the low-multilinear-rank tensor completion problem, a fundamental task in tensor-related applications. By exploiting the manifold structure inherent to the fixed-multilinear-rank tensor set, we introduce a simple yet highly effective preconditioned Riemannian metric and propose the Preconditioned Riemannian Gradient Descent (PRGD) algorithm. Compared to the standard Riemannian Gradient Descent (RGD), PRGD achieves faster convergence while maintaining the same order of per-iteration computational complexity. Theoretically, we provide the recovery guarantee for PRGD under near-optimal sampling complexity. Numerical results highlight the efficiency of PRGD, outperforming state-of-the-art methods on both synthetic data and real-world video inpainting tasks.

In the field of medical imaging driven by the Internet of Things (IoT), with the rapid growth of the number of medical devices and the widespread application of edge computing (EC) technology, efficient collaborative computing on resource-constrained end medical devices has become the key to improving diagnostic efficiency, thereby bringing a more patient-friendly diagnosis and treatment experience. Computed tomography perfusion (CTP) images play an irreplaceable role in the assessment of brain tissue ischemia in patients with acute ischemic stroke (AIS), with high diagnostic accuracy in identifying ischemic lesions and distinguishing infarction from penumbra, but it has the disadvantages of high radiation dose and high cost. To this end, we propose a CTP image synthesis framework based on edge-end collaboration (EESyn-CTP), which aims to use non-contrast CT (NCCT), CT angiography (CTA), and delayed CTA (CTA+8s) images to synthesize CTP images with arbitrary time to optimize AIS diagnosis. The framework consists of two stages: the pre-training stage on the edge server and the fine-tuning stage on the end device. Specifically, we first deploy a temporal residual generative network, t-UNet, on the edge server for pre-training. This process utilizes multiple CTP images, which share similar perfusion features with CTA, CTA+8s, and NCCT images, to effectively learn the gap in perfusion information between the inputs and outputs. Subsequently, the pre-trained t-UNet model parameters are frozen and broadcast to the edge medical device. A UNet adapter is introduced before the model, and fine-tuning is performed on the adapter weights using real NCCT, CTA, and CTA+8s images as input. This approach facilitates the synthesis of CTP images at arbitrary time points. Finally, experiments on an internal data set showed that the quality of Syn-CTP images synthesized by the EESyn-CTP framework is comparable to that of real CTP images and significantly reduces computation latency and energy overhead.

Abstract. Finding a transformation between two unknown probability distributions from finite samples is crucial for modeling complex data distributions and performing tasks such as sample generation, domain adaptation, and statistical inference. One powerful framework for such transformations is normalizing flow, which transforms an unknown distribution into a standard normal distribution using an invertible network. In this paper, we introduce a novel model called SyMOT-Flow, which trains an invertible transformation by minimizing the symmetric maximum mean discrepancy between samples from two unknown distributions, and an optimal transport cost is incorporated as regularization to obtain a short-distance and interpretable transformation. The resulting transformation leads to more stable and accurate sample generation. Several theoretical results are established for the proposed model, and its effectiveness is validated with illustrative low-dimensional examples as well as high-dimensional bimodality medical image generation through the forward and reverse flows.

Abstract. The reconstruction of dynamic positron emission tomography (PET) images from noisy projection data is a significant but challenging problem. In this paper, we introduce an unsupervised learning approach, non-negative implicit neural representation factorization, based on low rank matrix factorization of unknown images and employing neural networks to represent both coefficients and bases. Mathematically, we demonstrate that if a sequence of dynamic PET images satisfies a generalized non-negative low-rank property, it can be decomposed into a set of non-negative continuous functions varying in the temporal-spatial domain. This bridges the well-established non-negative matrix factorization with continuous functions, and we propose using implicit neural representations to connect matrix with continuous functions. The neural network parameters are obtained by minimizing the KL divergence, with additional sparsity regularization on coefficients and bases. Extensive experiments on dynamic PET reconstruction with Poisson noise demonstrate the effectiveness of the proposed method compared to other methods while giving continuous representations for object’s detailed geometric features and regional concentration variation.

Black-box optimization problems, which are common in many real-world applications, require optimization through input-output interactions without access to internal workings. This often leads to significant computational resources being consumed for simulations. Bayesian Optimization (BO) and Surrogate-Assisted Evolutionary Algorithm (SAEA) are two widely used gradient-free optimization techniques employed to address such challenges. Both approaches follow a similar iterative procedure that relies on surrogate models to guide the search process. This paper aims to elucidate the similarities and differences in the utilization of model uncertainty between these two methods, as well as the impact of model inaccuracies on algorithmic performance. A novel model-assisted strategy is introduced, which utilizes unevaluated solutions to generate offspring, leveraging the population-based search capabilities of evolutionary algorithm to enhance the effectiveness of model-assisted optimization. Experimental results demonstrate that the proposed approach outperforms mainstream Bayesian optimization algorithms in terms of accuracy and efficiency.

Abstract. Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (normalizing flow-based unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pretrained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and nonasymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.

The performance of graph representation learning is affected by the quality of graph input. While existing research usually pursues a globally smoothed graph embedding, we believe the rarely observed anomalies are as well harmful to an accurate prediction. This work establishes a graph learning scheme that automatically detects (locally) corrupted feature attributes and recovers robust embedding for prediction tasks. The detection operation leverages a graph autoencoder, which does not make any assumptions about the distribution of the local corruptions. It pinpoints the positions of the anomalous node attributes in an unbiased mask matrix, where robust estimations are recovered with sparsity promoting regularizer. The optimizer approaches a new embedding that is sparse in the framelet domain and conditionally close to input observations. Extensive experiments are provided to validate our proposed model can recover a robust graph representation from black-box poisoning and achieve excellent performance.

In this paper we combine the stochastic variance reduced gradient (SVRG) method [R. Johnson and T. Zhang, in Advances in Neural Information Processing Systems 26, 2013, pp. 315--323] with the primal dual fixed point method (PDFP) proposed in [P. Chen, J. Huang, and X. Zhang, Inverse Problems, 29 (2013)] to minimize a sum of two convex functions, one of which is linearly composite. This type of problems typically arise in sparse signal and image reconstruction. The proposed SVRG-PDFP can be seen as a generalization of Prox-SVRG [L. Xiao and T. Zhang, SIAM J. Optim., 24 (2014), pp. 2057--2075] originally designed for the minimization of a sum of two convex functions. Based on some standard assumptions, we propose two variants, one for strongly convex objective functions and the other for the general convex case. Convergence analysis shows that the convergence rate of SVRG-PDFP is $\mathcal{O}(\frac{1}{k})$ (here $k$ is the iteration number) for the general convex objective function and linear for the strongly convex case. Numerical examples on machine learning and computerized tomography image reconstruction are provided to show the effectiveness of the algorithms.

We provide a complete framework for performing infinite dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second, we provide a dimension-independent Markov chain Monte Carlo algorithm, based on the preconditioned Crank--Nicolson Langevin method, in which we use a primal-dual scheme to compute the offset direction. Third, we give a method combining the model discrepancy method and maximum likelihood estimation to determine the regularization parameter in the hybrid prior. Finally we propose to use the obtained posterior distribution to detect artifacts in a recovered image. We provide an example to demonstrate the effectiveness of the proposed method.

Recent technical advances lead to the coupling of PET and MRI scanners, enabling one to acquire functional and anatomical data simultaneously. In this paper, we propose a tight frame based PET-MRI joint reconstruction model via the joint sparsity of tight frame coefficients. In addition, a nonconvex balanced approach is adopted to take the different regularities of PET and MRI images into account. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm is proposed, and the global convergence is present based on the Kurdyka--Łojasiewicz property. Finally, the numerical experiments show that our proposed models achieve better performance over the existing PET-MRI joint reconstruction models.

Ultrasound video segmentation is a challenging task due to low contrast, shadow effects, complex noise statistics, and the need for high precision and efficiency in real time applications such as operation navigation and therapy planning. In this paper, we propose a wavelet frame based video segmentation framework incorporating different noise statistics and sequential distance shape priors. The proposed individual frame nonconvex segmentation model is solved by a proximal alternating minimization algorithm, and the convergence of the scheme is established based on the recently proposed Kurdyka--Łojasiewicz property. The performance of the overall method is demonstrated through numerical results on two real ultrasound video data sets. The proposed method is shown to achieve better results compared to the related level sets models and edge indicator shape priors, in terms of both segmentation quality and computational time.

This work develops a material reconstruction method for spectral CT, namely Total Image Constrained Material Reconstruction (TICMR), to maximize the utility of projection data in terms of both spectral information and high signal-to-noise ratio (SNR). This is motivated by the following fact: when viewed as a spectrally-integrated measurement, the projection data can be used to reconstruct a total image without spectral information, which however has a relatively high SNR; when viewed as a spectrally-resolved measurement, the projection data can be utilized to reconstruct the material composition, which however has a relatively low SNR. The material reconstruction synergizes material decomposition and image reconstruction, i.e., the direct reconstruction of material compositions instead of a two-step procedure that first reconstructs images and then decomposes images. For material reconstruction with high SNR, we propose TICMR with nonlocal total variation (NLTV) regularization. That is, first we reconstruct a total image using spectrally-integrated measurement without spectral binning, and build the NLTV weights from this image that characterize nonlocal image features; then the NLTV weights are incorporated into a NLTV-based iterative material reconstruction scheme using spectrally-binned projection data, so that these weights serve as a high-SNR reference to regularize material reconstruction. Note that the nonlocal property of NLTV is essential for material reconstruction, since material compositions may have significant local intensity variations although their structural information is often similar. In terms of solution algorithm, TICMR is formulated as an iterative reconstruction method with the NLTV regularization, in which the nonlocal divergence is utilized based on the adjoint relationship. The alternating direction method of multipliers is developed to solve this sparsity optimization problem. The proposed TICMR method was validated using both simulated and experimental data. In comparison with FBP and total-variation-based iterative method, TICMR had improved image quality, e.g., contrast-to-noise ratio and spatial resolution.

Wavelet frames have been successfully applied to various image restoration problems, such as denoising, inpainting, and deblurring. However, they are rarely used in geometric applications, except for the recent work of [B. Dong, A. Chien, and Z. Shen, Commun. Math. Sci., 9 (2011), pp. 551--559; B. Dong and Z. Shen, in Proceedings of the SPIE, Vol. 8401, 2012, 840102]. Motivated by the theoretical connection between wavelet frame based and total variation based image restoration models recently established in [J.-F. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033--1089], we propose here a convex multiphase segmentation model based on wavelet frame transform. The proposed model allows us to automatically identify complex tubular structures, including blood vessels, leaf vein systems, etc. Numerical results show that our method can extract more details than existing variational methods especially when the image contains different scales of structures. The proposed method is parallelized, and its efficiency is further improved by a graphics processing unit implementation. In addition, we analyze the connection between solutions of the convexified model and the original binary constrained model.

We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by Zhu and Chan in [An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, CAM Report 08-34, UCLA, Los Angeles, CA, 2008] to a broader class of convex optimization problems. In addition, we survey several closely related methods and explain the connections to PDHG. We point out convergence results for a modified version of PDHG that has a similarly good empirical convergence rate for total variation (TV) minimization problems. We also prove a convergence result for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. We show how to interpret this special case as a projected averaged gradient method applied to the dual functional. We discuss the range of parameters for which these methods can be shown to converge. We also present some numerical comparisons of these algorithms applied to TV denoising, TV deblurring, and constrained $l_1$ minimization problems.

Bregman methods introduced in [S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, Multiscale Model. Simul., 4 (2005), pp. 460–489] to image processing are demonstrated to be an efficient optimization method for solving sparse reconstruction with convex functionals, such as the $\ell^1$ norm and total variation [W. Yin, S. Osher, D. Goldfarb, and J. Darbon, SIAM J. Imaging Sci., 1 (2008), pp. 143–168; T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323–343]. In particular, the efficiency of this method relies on the performance of inner solvers for the resulting subproblems. In this paper, we propose a general algorithm framework for inverse problem regularization with a single forward-backward operator splitting step [P. L. Combettes and V. R. Wajs, Multiscale Model. Simul., 4 (2005), pp. 1168–1200], which is used to solve the subproblems of the Bregman iteration. We prove that the proposed algorithm, namely, Bregmanized operator splitting (BOS), converges without fully solving the subproblems. Furthermore, we apply the BOS algorithm and a preconditioned one for solving inverse problems with nonlocal functionals. Our numerical results on deconvolution and compressive sensing illustrate the performance of nonlocal total variation regularization under the proposed algorithm framework, compared to other regularization techniques such as the standard total variation method and the wavelet-based regularization method.