Search

We consider the problem of minimizing a convex separable objective (as a separable sum of two proper closed convex functions $f$ and $g$) over a linear coupling constraint. We assume that $f$ can be decomposed as the sum of a smooth part having H\"older continuous gradient (with exponent $\mu\in(0,1]$) and a nonsmooth part that admits efficient proximal mapping computations, while $g$ can be decomposed as the sum of a smooth part having H\"older continuous gradient (with exponent $\nu\in(0,1]$) and a nonsmooth part that admits efficient linear oracles. Motivated by the recent work [41], we propose a single-loop variant of the standard penalty method, which we call a single-loop proximal-conditional-gradient penalty method ($proxCG_{1\ell}^{pen}$), for this problem. In each iteration, we successively perform one proximal-gradient step involving $f$ and one conditional-gradient step involving $g$ on the quadratic penalty function, followed by an update of the penalty parameter. We present explicit rules for updating the penalty parameter and the stepsize in the conditional-gradient step. Under a standard constraint qualification and domain boundedness assumption, we show that the objective value deviations (from the optimal value) along the sequence generated decay in the order of $t^{-\min\{\mu,\nu,1/2\}}$ with the associated feasibility violations decaying in the order of $t^{-1/2}$. Moreover, if the nonsmooth parts are indicator functions and the extended objective is a Kurdyka-Lojasiewicz function with exponent $\alpha\in [0,1)$, then the distances to the optimal solution set along the sequence generated by $proxCG_{1\ell}^{pen}$ decay asymptotically at a rate of $t^{-(1-\alpha)\min\{\mu,\nu,1/2\}}$. Finally, we illustrate numerically the convergence behavior of $proxCG_{1\ell}^{pen}$ on minimizing the $\ell_1$ norm subject to a residual error measured by $\ell_p$ norm, $p\in(1,2]$.

Image environments and noisy labels hinder deep learning-based inference models in structural damage detection. Post-detection, there is the challenge of reliance on manual assessments of detected damages. As a result, Guided-DetNet, characterized by Generative Attention Module (GAM), Hierarchical Elimination Algorithm (HEA), and Volumetric Contour Visual Assessment (VCVA), is proposed to mitigate complex image environments, noisy labeling, and post-detection manual assessment of structural damages. GAM leverages cross-horizontal and cross-vertical patch merging and cross foreground-background feature fusion to generate varied features to mitigate complex image environments. HEA addresses noisy labeling using hierarchical relationships among classes to refine instances given an image by eliminating unlikely class categories. VCVA assesses the severity of detected damages via volumetric representation and quantification leveraging the Dirac delta distribution. A comprehensive quantitative study, two robustness tests, and an application scenario based on the PEER Hub Image-Net dataset substantiate Guided-DetNet's promising performances. Guided-DetNet outperformed the best-compared models in a triple classification task by a difference of not less than 3% and not less than 2% in a dual detection task under varying metrics.

Accurate tooth volume segmentation is a prerequisite for computer-aided dental analysis. Deep learning-based tooth segmentation methods have achieved satisfying performances but require a large quantity of tooth data with ground truth. The dental data publicly available is limited meaning the existing methods can not be reproduced, evaluated and applied in clinical practice. In this paper, we establish a 3D dental CBCT dataset CTooth+, with 22 fully annotated volumes and 146 unlabeled volumes. We further evaluate several state-of-the-art tooth volume segmentation strategies based on fully-supervised learning, semi-supervised learning and active learning, and define the performance principles. This work provides a new benchmark for the tooth volume segmentation task, and the experiment can serve as the baseline for future AI-based dental imaging research and clinical application development.

3D tooth segmentation is a prerequisite for computer-aided dental diagnosis and treatment. However, segmenting all tooth regions manually is subjective and time-consuming. Recently, deep learning-based segmentation methods produce convincing results and reduce manual annotation efforts, but it requires a large quantity of ground truth for training. To our knowledge, there are few tooth data available for the 3D segmentation study. In this paper, we establish a fully annotated cone beam computed tomography dataset CTooth with tooth gold standard. This dataset contains 22 volumes (7363 slices) with fine tooth labels annotated by experienced radiographic interpreters. To ensure a relative even data sampling distribution, data variance is included in the CTooth including missing teeth and dental restoration. Several state-of-the-art segmentation methods are evaluated on this dataset. Afterwards, we further summarise and apply a series of 3D attention-based Unet variants for segmenting tooth volumes. This work provides a new benchmark for the tooth volume segmentation task. Experimental evidence proves that attention modules of the 3D UNet structure boost responses in tooth areas and inhibit the influence of background and noise. The best performance is achieved by 3D Unet with SKNet attention module, of 88.04 \% Dice and 78.71 \% IOU, respectively. The attention-based Unet framework outperforms other state-of-the-art methods on the CTooth dataset. The codebase and dataset are released.

The Frank-Wolfe (FW) method, which implements efficient linear oracles that minimize linear approximations of the objective function over a fixed compact convex set, has recently received much attention in the optimization and machine learning literature. In this paper, we propose a new FW-type method for minimizing a smooth function over a compact set defined as the level set of a single difference-of-convex function, based on new generalized linear-optimization oracles (LO). We show that these LOs can be computed efficiently with closed-form solutions in some important optimization models that arise in compressed sensing and machine learning. In addition, under a mild strict feasibility condition, we establish the subsequential convergence of our nonconvex FW-type method. Since the feasible region of our generalized LO typically changes from iteration to iteration, our convergence analysis is completely different from those existing works in the literature on FW-type methods that deal with fixed feasible regions among subproblems. Finally, motivated by the away steps for accelerating FW-type methods for convex problems, we further design an away-step oracle to supplement our nonconvex FW-type method, and establish subsequential convergence of this variant. Numerical results on the matrix completion problem with standard datasets are presented to demonstrate the efficiency of the proposed FW-type method and its away-step variant., Comment: We updated grant information and fixed some minor typos in Section 8

Trust-region (TR) type method, based on a quadratic model such as the trust-region subproblem (TRS) and $ p $-regularization subproblem ($p$RS), is arguably one of the most successful methods for unconstrained minimization. In this paper, we study a general regularized subproblem (named $ \rho $RS), which covers TRS and $p$RS as special cases. We derive a strong duality theorem for $ \rho $RS, and also its necessary and sufficient optimality condition under general assumptions on the regularization term. We then define the Rendl-Wolkowicz (RW) dual problem of $ \rho $RS, which is a maximization problem whose objective function is concave, and differentiable except possibly at two points. It is worth pointing out that our definition is based on an alternative derivation of the RW-dual problem for TRS. Then we propose an eigensolver-based algorithm for solving the RW-dual problem of $ \rho $RS. The algorithm is carried out by finding the smallest eigenvalue and its unit eigenvector of a certain matrix in each iteration. Finally, we present numerical results on randomly generated $p$RS's, and on a new class of regularized problem that combines TRS and $p$RS, to illustrate our algorithm.

Recently, in a series of papers [32,38,39,41], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing $\ell_1$/$\ell_2$ subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [32] satisfies the Kurdyka-Lojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [39, Eq. 11] under mild assumptions. We next extend the $\ell_1$/$\ell_2$ model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [37,44], and extend the algorithm in [39, Eq. 11] by incorporating moving-balls-approximation techniques [4] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions, and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by $ \ell_2 $ norm or Lorentzian norm via solving the corresponding $\ell_1$/$\ell_2$ models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy.

The split feasibility problem is to find an element in the intersection of a closed set $C$ and the linear preimage of another closed set $D$, assuming the projections onto $C$ and $D$ are easy to compute. This class of problems arises naturally in many contemporary applications such as compressed sensing. While the sets $C$ and $D$ are typically assumed to be convex in the literature, in this paper, we allow both sets to be possibly nonconvex. We observe that, in this setting, the split feasibility problem can be formulated as an optimization problem with a difference-of-convex objective so that standard majorization-minimization type algorithms can be applied. Here we focus on the nonmonotone proximal gradient algorithm with majorization studied in [15, Appendix A]. We show that, when this algorithm is applied to a split feasibility problem, the sequence generated clusters at a stationary point of the problem under mild assumptions. We also study local convergence property of the sequence under suitable assumptions on the closed sets involved. Finally, we perform numerical experiments to illustrate the efficiency of our approach on solving split feasibility problems that arise in completely positive matrix factorization, (uniformly) sparse matrix factorization, and outlier detection., Comment: Published in JOGO. In this ArXiv version, 1. We fix an unfortunate error in Lemma 4 and the paragraph after it: the matrices involved should have rank r instead of being full rank; 2. An additional compactness condition is needed for the uniform KL exponent in Proposition 5

In blockchain systems, especially cryptographic currencies such as Bitcoin, the double-spending and Byzantine-general-like problem are solved by reaching consensus protocols among all nodes. The state-of-the-art protocols include Proof-of-Work, Proof-of-Stake and Delegated-Proof-of-Stake. Proof-of-Work urges nodes to prove their computing power measured in hash rate in a crypto-puzzle solving competition. The other two take into account the amount of stake of each nodes and even design a vote in Delegated-Proof-of-Stake. However, these frameworks have several drawbacks, such as consuming a large number of electricity, leading the whole blockchain to a centralized system and so on. In this paper, we propose the conceptual framework, fundamental theory and research methodology, based on artificial intelligence technology that exploits nearly complementary information of each nodes. And we designed a particular convolutional neural network and a dynamic threshold, which obtained the super nodes and the random nodes, to reach the consensus. Experimental results demonstrate that our framework combines the advantages of Proof-of-Work, Proof-of-Stake and Delegated-Proof-of-Stake by avoiding complicated hash operation and monopoly. Furthermore, it compares favorably to the three state-of-the-art consensus frameworks, in terms of security and the speed of transaction confirmation.