Critical Points Layer (CPL)
While several convolution-like operators have recently been proposed for extracting features out of point clouds, down-sampling an unordered point cloud in a deep neural network has not been rigorously studied. Existing methods down-sample the points regardless of their importance for the output. As a result, some important points in the point cloud may be removed, while less valuable points may be passed to the next layers. In contrast, adaptive down-sampling methods sample the points by taking into account the importance of each point, which varies based on the application, task and training data. In this paper, we propose a permutation-invariant learning-based adaptive down-sampling layer, called Critical Points Layer (CPL), which reduces the number of points in an unordered point cloud while retaining the important points. Unlike most graph-based point cloud down-sampling methods that use $k$-NN search algorithm to find the neighbouring points, CPL is a global down-sampling method, rendering it computationally very efficient. The proposed layer can be used along with any graph-based point cloud convolution layer to form a convolutional neural network, dubbed CP-Net in this paper. We introduce a CP-Net for $3$D object classification that achieves the best accuracy for the ModelNet$40$ dataset among point cloud-based methods, which validates the effectiveness of the CPL. …

Complex Network Classifier (CNC)
Classifying large scale networks into several categories and distinguishing them according to their fine structures is of great importance with several applications in real life. However, most studies of complex networks focus on properties of a single network but seldom on classification, clustering, and comparison between different networks, in which the network is treated as a whole. Due to the non-Euclidean properties of the data, conventional methods can hardly be applied on networks directly. In this paper, we propose a novel framework of complex network classifier (CNC) by integrating network embedding and convolutional neural network to tackle the problem of network classification. By training the classifiers on synthetic complex network data and real international trade network data, we show CNC can not only classify networks in a high accuracy and robustness, it can also extract the features of the networks automatically. …

Differentiable Linearized ADMM (D-LADMM)
Recently, a number of learning-based optimization methods that combine data-driven architectures with the classical optimization algorithms have been proposed and explored, showing superior empirical performance in solving various ill-posed inverse problems, but there is still a scarcity of rigorous analysis about the convergence behaviors of learning-based optimization. In particular, most existing analyses are specific to unconstrained problems but cannot apply to the more general cases where some variables of interest are subject to certain constraints. In this paper, we propose Differentiable Linearized ADMM (D-LADMM) for solving the problems with linear constraints. Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is obtained by firstly introducing some learnable weights in the classical Linearized ADMM algorithm and then generalizing the proximal operator to some learnable activation function. Notably, we rigorously prove that there exist a set of learnable parameters for D-LADMM to generate globally converged solutions, and we show that those desired parameters can be attained by training D-LADMM in a proper way. To the best of our knowledge, we are the first to provide the convergence analysis for the learning-based optimization method on constrained problems. …

Yakmo
Yakmo implements robust, efficient k-means clustering with triangular inequality and smart initialization , while supporting alternative clustering outputs. The use of the triangular inequality allows k-means to skip unnecessary distance calculations, while the smart initialization by randomized seeding (k-means++) not only improves solution accuracy but also accelerates the convergence of the algorithm. In addition, you can obtain alternative clusterings via orthogonalization . …