De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem

The Weber location problem is widely used in several artificial intelligence scenarios. However, the gradient of the objective does not exist at a considerable set of singular points. Recently, a de-singularity subgradient method has been proposed to fix this problem, but it can only handle the $q$-th-powered $\ell_2$-norm case ($1\leqslant q<2$), which has only finite singular points. In this paper, we further establish the de-singularity subgradient for the $q$-th-powered $\ell_p$-norm case with $1\leqslant q\leqslant p$ and $1\leqslant p<2$, which includes all the rest unsolved situations in this problem. This is a challenging task because the singular set is a continuum. The geometry of the objective function is also complicated so that the characterizations of the subgradients, minimum and descent direction are very difficult. We develop a $q$-th-powered $\ell_p$-norm Weiszfeld Algorithm without Singularity ($q$P$p$NWAWS) for this problem, which ensures convergence and the descent property of the objective function. Extensive experiments on six real-world data sets demonstrate that $q$P$p$NWAWS successfully solves the singularity problem and achieves a linear computational convergence rate in practical scenarios.

TDCNet: Transparent Objects Depth Completion with CNN-Transformer Dual-Branch Parallel Network

The sensing and manipulation of transparent objects present a critical challenge in industrial and laboratory robotics. Conventional sensors face challenges in obtaining the full depth of transparent objects due to the refraction and reflection of light on their surfaces and their lack of visible texture. Previous research has attempted to obtain complete depth maps of transparent objects from RGB and damaged depth maps (collected by depth sensor) using deep learning models. However, existing methods fail to fully utilize the original depth map, resulting in limited accuracy for deep completion. To solve this problem, we propose TDCNet, a novel dual-branch CNN-Transformer parallel network for transparent object depth completion. The proposed framework consists of two different branches: one extracts features from partial depth maps, while the other processes RGB-D images. Experimental results demonstrate that our model achieves state-of-the-art performance across multiple public datasets. Our code and the pre-trained model are publicly available at https://github.com/XianghuiFan/TDCNet.

A De-singularity Subgradient Approach for the Extended Weber Location Problem

The extended Weber location problem is a classical optimization problem that has inspired some new works in several machine learning scenarios recently. However, most existing algorithms may get stuck due to the singularity at the data points when the power of the cost function $1\leqslant q<2$, such as the widely-used iterative Weiszfeld approach. In this paper, we establish a de-singularity subgradient approach for this problem. We also provide a complete proof of convergence which has fixed some incomplete statements of the proofs for some previous Weiszfeld algorithms. Moreover, we deduce a new theoretical result of superlinear convergence for the iteration sequence in a special case where the minimum point is a singular point. We conduct extensive experiments in a real-world machine learning scenario to show that the proposed approach solves the singularity problem, produces the same results as in the non-singularity cases, and shows a reasonable rate of linear convergence. The results also indicate that the $q$-th power case ($1<q<2$) is more advantageous than the $1$-st power case and the $2$-nd power case in some situations. Hence the de-singularity subgradient approach is beneficial to advancing both theory and practice for the extended Weber location problem.

Select Good Regions for Deblurring based on Convolutional Neural Networks

The goal of blind image deblurring is to recover sharp image from one input blurred image with an unknown blur kernel. Most of image deblurring approaches focus on developing image priors, however, there is not enough attention to the influence of image details and structures on the blur kernel estimation. What is the useful image structure and how to choose a good deblurring region? In this work, we propose a deep neural network model method for selecting good regions to estimate blur kernel. First we construct image patches with labels and train a deep neural networks, then the learned model is applied to determine which region of the image is most suitable to deblur. Experimental results illustrate that the proposed approach is effective, and could be able to select good regions for image deblurring.