# Graph Interpolating Activation Improves Both Natural and Robust Accuracies in Data-Efficient Deep Learning

## TL;DR

This paper introduces a graph Laplacian-based interpolating activation function for deep neural networks, enhancing accuracy and robustness, especially in data-efficient and semi-supervised learning scenarios, by integrating manifold learning principles.

## Contribution

It proposes a novel graph interpolating activation replacing softmax, improving data efficiency, robustness, and semi-supervised learning in deep neural networks.

## Key findings

- Improves natural and robust accuracy on adversarial images.
- Enhances data-efficient learning with high-capacity DNNs.
- Demonstrates effectiveness in semi-supervised learning.

## Abstract

Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace-Beltrami equation on a high dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation integrates the advantages of both deep learning and manifold learning. Compared to the conventional DNNs with the softmax function as output activation, the new framework demonstrates the following major advantages: First, it is better applicable to data-efficient learning in which we train high capacity DNNs without using a large number of training data. Second, it remarkably improves both natural accuracy on the clean images and robust accuracy on the adversarial images crafted by both white-box and black-box adversarial attacks. Third, it is a natural choice for semi-supervised learning. For reproducibility, the code is available at \url{https://github.com/BaoWangMath/DNN-DataDependentActivation}.