Mapper Based Classifier

--T opological data analysis aims to extract topological quantities from data, which tend to focus on the broader global structure of the data rather than local information. The Mapper method, specifically, generalizes clustering methods to identify significant global mathematical structures, which are out of reach of many other approaches. We propose a classifier based on applying the Mapper algorithm to data projected onto a latent space. We obtain the latent space by using PCA or autoencoders. Notably, a classifier based on the Mapper method is immune to any gradient based attack, and improves robustness over traditional CNNs (convolutional neural networks). We report theoretical justification and some numerical experiments that confirm our claims. I NTRODUCTION Deep neural networks [1], [2] are well known to be not robust with respect to input image perturbations, which are designed by adding to images perturbations that are typically non-perceptible by humans [3]-[5]. In this paper we explore opportunities for combining deep learning techniques with a well known topological data analysis (TDA) algorithm - the Mapper algorithm [6], which we use to create classifiers with improved robustness. First, the training data is projected onto a latent space. The latent space in the simplest variant is constructed using PCA components, and we also use nonlinear projections by utilizing various autoencoders [1], [7]-[9]. Then, a discrete graph representation (Mapper) is assigned to the training data projected onto the latent space .