Abstract: In this paper, we present a novel classification method called deep diffusion-invariant Wasserstein distributional classification (DeepWDC). DeepWDC represents input data and labels as probability measures to address severe perturbations in input data. It can output the optimal label measure in terms of diffusion invariance, where the label measure is stationary over time and becomes equivalent to a Gaussian measure. Furthermore, DeepWDC minimizes the 2-Wasserstein distance between the optimal label measure and Gaussian measure, which reduces the Wasserstein uncertainty. Experimental results demonstrate that DeepWDC can substantially enhance the accuracy of several baseline deterministic classification methods and outperforms state-of-the-art-methods on 2D and 3D data containing various types of perturbations (e.g., rotations, impulse noise, and down-scaling).