Batch Normalization (BatchNorm) is effective for improving the performance and accelerating the t... more Batch Normalization (BatchNorm) is effective for improving the performance and accelerating the training of deep neural networks. However, it has also shown to be a cause of adversarial vulnerability, i.e., networks without it are more robust to adversarial attacks. In this paper, we investigate how BatchNorm causes this vulnerability and proposed new normalization that is robust to adversarial attacks. We first observe that adversarial images tend to shift the distribution of BatchNorm input, and this shift makes train-time estimated population statistics inaccurate. We hypothesize that these inaccurate statistics make models with BatchNorm more vulnerable to adversarial attacks. We prove our hypothesis by replacing train-time estimated statistics with statistics calculated from the inference-time batch. We found that the adversarial vulnerability of BatchNorm disappears if we use these statistics. However, without estimated batch statistics, we can not use BatchNorm in the practic...
International Journal of Computer Mathematics, 2007
The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. ... more The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. This method is obtained from M. Javidi's method (Appl. Math. Comput. 193 (2007) 360-365), which is third-order. The convergence order of new method is established to six and the e¢ ciency index is 1.5651. The Proposed method is compared with the second, third and sixth order methods. Some numerical test problems are given to show the accuracy and fast convergence of the proposed method.
In this paper, we present a new modified variant of super-Halley method for solving non-linear eq... more In this paper, we present a new modified variant of super-Halley method for solving non-linear equations. Numerical results show that the method has definite practical utility.
International Journal of Computer Mathematics, 2007
The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. ... more The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. This method is obtained from M. Javidi's method (Appl. Math. Comput. 193 (2007) 360-365), which is third-order. The convergence order of new method is established to six and the e¢ ciency index is 1.5651. The Proposed method is compared with the second, third and sixth order methods. Some numerical test problems are given to show the accuracy and fast convergence of the proposed method.
In this paper, we present a new modified variant of super-Halley method for solving non-linear eq... more In this paper, we present a new modified variant of super-Halley method for solving non-linear equations. Numerical results show that the method has definite practical utility.
Uploads
Papers by Muhammad Awais