This letter proposes a gradient-enhanced softmax supervisor for face recognition (FR) based on a deep convolutional neural network (DCNN). The proposed supervisor conducts the constant-normalized cosine to obtain the score for each class using a combination of the intra-class score and the soft maximum of the inter-class scores as the objective function. This mitigates the vanishing gradient problem in the conventional softmax classifier. The experiments on the public Labeled Faces in the Wild (LFW) database denote that the proposed supervisor achieves better results when compared with those achieved using the current state-of-the-art softmax-based approaches for FR.

As a with-carry analog (based on modular arithmetic) of the usual Walsh-Hadamard transform (WHT), arithmetic Walsh transform (AWT) has been used to obtain analogs of some properties of Boolean functions which are important in the design and analysis of cryptosystems. The existence of nonzero linear structure of Boolean functions is an important criterion to measure the weakness of these functions in their cryptographic applications. In this paper, we find more analogs of linear structures of Boolean functions from AWT. For some classes of n-variable Boolean functions f, we find necessary and sufficient conditions for the existence of an invariant linear structure and a complementary linear structure 1n of f. We abstract out a sectionally linear relationship between AWT and WHT of n-variable balanced Boolean functions f with linear structure 1n. This result show that AWT can characterize cryptographic properties of these functions as long as WHT can. In addition, for a diagonal Boolean function f, a recent result by Carlet and Klapper says that the AWT of f can be expressed in terms of the AWT of a diagonal Boolean function of algebraic degree at most 3 in a larger number of variables. We provide for the result a complete and more modular proof which works for both even and odd weights (of the parameter c in the Corollary 19 by Carlet and Klapper (DCC 73(2): 299-318, 2014).