The step function is one of the simplest and most natural activation functions for deep neural networks (DNNs). As it counts 1 for positive variables and 0 for others, its intrinsic characteristics (e.g., discontinuity and no viable information of subgradients) impede its development for several decades. Even if there is an impressive body of work on designing DNNs with continuous activation functions that can be deemed as surrogates of the step function, it is still in the possession of some advantageous properties, such as complete robustness to outliers and being capable of attaining the best learning-theoretic guarantee of predictive accuracy. Hence, in this paper, we aim to train DNNs with the step function used as an activation function (dubbed as 0/1 DNNs). We first reformulate 0/1 DNNs as an unconstrained optimization problem and then solve it by a block coordinate descend (BCD) method. Moreover, we acquire closed-form solutions for sub-problems of BCD as well as its convergence properties. Furthermore, we also integrate $\ell_{2,0}$-regularization into 0/1 DNN to accelerate the training process and compress the network scale. As a result, the proposed algorithm has a high performance on classifying MNIST and Fashion-MNIST datasets. As a result, the proposed algorithm has a desirable performance on classifying MNIST, FashionMNIST, Cifar10, and Cifar100 datasets.

JOURNAL OF L A T EX CLASS FILES, VOL., NO., 1 Single V ersus Union: Nonparallel Support V ector Machine Frameworks Chun-Na Li, Y uan-Hai Shao, Huajun Wang, Y u-Ting Zhao, Ling-Wei Huang, Naihua Xiu and Nai-Y ang Deng Abstract --Considering the classification problem, we summarize the nonparallel support vector machines with the nonparallel hyperplanes to two types of frameworks. It solves a series of small optimization problems to obtain a series of hyperplanes, but is hard to measure the loss of each sample. The other type constructs all the hyperplanes simultaneously, and it solves one big optimization problem with the ascertained loss of each sample. We give the characteristics of each framework and compare them carefully. In addition, based on the second framework, we construct a max-min distance-based nonparallel support vector machine for multiclass classification problem, called NSVM. Experimental results on benchmark data sets and human face databases show the advantages of our NSVM. I NTRODUCTION F OR binary classification problem, the generalized eigenvalue proximal support vector machine (GEPSVM) was proposed by Mangasarian and Wild [1] in 2006, which is the first nonparallel support vector machine. It aims at generating two nonparallel hyperplanes such that each hyperplane is closer to its class and as far as possible from the other class. GEPSVM is effective, particularly when dealing with the "Xor"-type data [1]. This leads to extensive studies on nonparallel support vector machines (NSVMs) [2]-[5].

This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices. We first benefit from a convex optimization which develops $l_1$-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property. For the proposed estimator, we then prove that with large probability, the Frobenious norm of the estimation rate can be of order $O(\sqrt{s(\log{r})/n})$ under a mild case, where $s$ and $r$ denote the number of sparse entries and the rank of the population covariance respectively, $n$ notes the sample capacity. Finally an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem, and meantime merits of the approach are also illustrated by practicing numerical simulations.