A Customized Augmented Lagrangian Method for Block-Structured Integer Programming.

Integer programming with block structures has received considerable attention recently and is widely used in many practical applications such as train timetabling and vehicle routing problems. It is known to be NP-hard due to the presence of integer variables. We define a novel augmented Lagrangian function by directly penalizing the inequality constraints and establish the strong duality between the primal problem and the augmented Lagrangian dual problem. Then, a customized augmented Lagrangian method is proposed to address the block-structures. In particular, the minimization of the augmented Lagrangian function is decomposed into multiple subproblems by decoupling the linking constraints and these subproblems can be efficiently solved using the block coordinate descent method. We also establish the convergence property of the proposed method. To make the algorithm more practical, we further introduce several refinement techniques to identify high-quality feasible solutions. Numerical experiments on a few interesting scenarios show that our proposed algorithm often achieves a satisfactory solution and is quite effective.

Predicting sequenced dental treatment plans from electronic dental records using deep learning.

BACKGROUND: Designing appropriate clinical dental treatment plans is an urgent need because a growing number of dental patients are suffering from partial edentulism with the population getting older. OBJECTIVES: The aim of this study is to predict sequential treatment plans from electronic dental records. METHODS: We construct a clinical decision support model, MultiTP, explores the unique topology of teeth information and the variation of complicated treatments, integrates deep learning models (convolutional neural network and recurrent neural network) adaptively, and embeds the attention mechanism to produce optimal treatment plans. RESULTS: MultiTP shows its promising performance with an AUC of 0.9079 and an F score of 0.8472 over five treatment plans. The interpretability analysis also indicates its capability in mining clinical knowledge from the textual data. CONCLUSIONS: MultiTP's novel problem formulation, neural network framework, and interpretability analysis techniques allow for broad applications of deep learning in dental healthcare, providing valuable support for predicting dental treatment plans in the clinic and benefiting dental patients. CLINICAL IMPLICATIONS: The MultiTP is an efficient tool that can be implemented in clinical practice and integrated into the existing EDR system. By predicting treatment plans for partial edentulism, the model will help dentists improve their clinical decisions.

An Efficient Fisher Matrix Approximation Method for Large-Scale Neural Network Optimization.

Although the shapes of the parameters are not crucial for designing first-order optimization methods in large scale empirical risk minimization problems, they have important impact on the size of the matrix to be inverted when developing second-order type methods. In this article, we propose an efficient and novel second-order method based on the parameters in the real matrix space [Formula: see text] and a matrix-product approximate Fisher matrix (MatFisher) by using the products of gradients. The size of the matrix to be inverted is much smaller than that of the Fisher information matrix in the real vector space [Formula: see text]. Moreover, by utilizing the matrix delayed update and the block diagonal approximation techniques, the computational cost can be controlled and is comparable with first-order methods. A global convergence and a superlinear local convergence analysis are established under mild conditions. Numerical results on image classification with ResNet50, quantum chemistry modeling with SchNet, and data-driven partial differential equations solution with PINN illustrate that our method is quite competitive to the state-of-the-art methods.

Pure State v-Representability of Density Matrix Embedding Theory.

Density matrix embedding theory (DMET) formally requires the matching of density matrix blocks obtained from high-level and low-level theories, but this is sometimes not achievable in practical calculations. In such a case, the global band gap of the low-level theory vanishes, and this can require additional numerical considerations. We find that both the violation of the exact matching condition and the vanishing low-level gap are related to the assumption that the high-level density matrix blocks are noninteracting pure-state v-representable (NI-PS-V), which assumes that the low-level density matrix is constructed following the Aufbau principle. To relax the NI-PS-V condition, we develop an augmented Lagrangian method to match the density matrix blocks without referring to the Aufbau principle. Numerical results for the 2D Hubbard and hydrogen model systems indicate that, in some challenging scenarios, the relaxation of the Aufbau principle directly leads to exact matching of the density matrix blocks, which also yields improved accuracy.

Clinical decision support model for tooth extraction therapy derived from electronic dental records.

STATEMENT OF PROBLEM: Tooth extraction therapy serves as a key initial step in many prosthodontic treatment plans. Dentists must make an appropriate decision on the tooth extraction therapy considering multiple determinants and whether a clinical decision support (CDS) model might help. PURPOSE: The purpose of this retrospective records study was to construct a CDS model to predict tooth extraction therapy in clinical situations by using electronic dental records (EDRs). MATERIAL AND METHODS: The cohort involved 4135 deidentified EDRs of 3559 patients from the database of a prosthodontics department. Knowledge-based algorithms were first proposed to convert raw data from EDRs into structured data for feature extraction. Redundant features were filtered by a recursive feature-elimination method. The tooth extraction problem was then modeled alternatively as a binary or triple classification problem to be solved by 5 machine learning algorithms. Five machine learning algorithms within each model were compared, as well as the efficiency between 2 models. In addition, the proposed CDS was verified by 2 prosthodontists. RESULTS: The triple classification model outperformed the binary model with the F1 score of the Extreme Gradient Boost (XGBoost) algorithm as 0.856 and 0.847, respectively. The XGBoost outperformed the other 4 algorithms. The accuracy, precision, and recall of the XGBoost algorithm were 0.962, 0.865, and 0.830 in the binary classification and 0.924, 0.879, and 0.836 in the triple classification, respectively. The performance of the 2 prosthodontists was inferior to the models. CONCLUSIONS: The CDS model for tooth extraction therapy achieved high performance in terms of decision-making derived from EDRs.

Orientation Determination of Cryo-EM Images Using Least Unsquared Deviations.

A major challenge in single particle reconstruction from cryo-electron microscopy is to establish a reliable ab initio three-dimensional model using two-dimensional projection images with unknown orientations. Common-lines-based methods estimate the orientations without additional geometric information. However, such methods fail when the detection rate of common-lines is too low due to the high level of noise in the images. An approximation to the least squares global self-consistency error was obtained in [A. Singer and Y. Shkolnisky, SIAM J. Imaging Sci., 4 (2011), pp. 543-572] using convex relaxation by semidefinite programming. In this paper we introduce a more robust global self-consistency error and show that the corresponding optimization problem can be solved via semidefinite relaxation. In order to prevent artificial clustering of the estimated viewing directions, we further introduce a spectral norm term that is added as a constraint or as a regularization term to the relaxed minimization problem. The resulting problems are solved using either the alternating direction method of multipliers or an iteratively reweighted least squares procedure. Numerical experiments with both simulated and real images demonstrate that the proposed methods significantly reduce the orientation estimation error when the detection rate of common-lines is low.

Direct fast method for time-limited signal reconstruction.

We consider reconstruction of signals by a direct method for the solution of the discrete Fourier system. We note that the reconstruction of a time-limited signal can be simply realized by using only either the real part or the imaginary part of the discrete Fourier transform (DFT) matrix. Therefore, based on the study of the special structure of the real and imaginary parts of the discrete Fourier matrix, we propose a fast direct method for the signal reconstruction problem, which utilizes the numerically truncated singular value decomposition. The method enables us to recover the original signal in a stable way from the frequency information, which may be corrupted by noise and/or some missing data. The classical inverse Fourier transform cannot be applied directly in the latter situation. The pivotal point of the reconstruction is the explicit computation of the singular value decomposition of the real part of the DFT for any order. Numerical experiments for 1D and 2D signal reconstruction and image restoration are given.