Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. To make the R-CDT invariant under arbitrary affine transformations, a two-step normalization of the R-CDT has been proposed in our earlier works. The aim of this paper is twofold. First, we propose a family of generalized normalizations to enhance flexibility for applications. Second, we study multi-dimensional and non-Euclidean settings by making use of generalized Radon transforms. We prove that our novel feature representations are invariant under certain transformations and allow for linear separation in feature space. Our theoretical results are supported by numerical experiments based on 2d images, 3d shapes and 3d rotation matrices, showing near perfect classification accuracies and clustering results.

Unsupervised Ground Metric Learning

Data classification without access to labeled samples remains a challenging problem. It usually depends on an appropriately chosen distance between features, a topic addressed in metric learning. Recently, Huizing, Cantini and Peyr\'e proposed to simultaneously learn optimal transport (OT) cost matrices between samples and features of the dataset. This leads to the task of finding positive eigenvectors of a certain nonlinear function that maps cost matrices to OT distances. Having this basic idea in mind, we consider both the algorithmic and the modeling part of unsupervised metric learning. First, we examine appropriate algorithms and their convergence. In particular, we propose to use the stochastic random function iteration algorithm and prove that it converges linearly for our setting, although our operators are not paracontractive as it was required for convergence so far. Second, we ask the natural question if the OT distance can be replaced by other distances. We show how Mahalanobis-like distances fit into our considerations. Further, we examine an approach via graph Laplacians. In contrast to the previous settings, we have just to deal with linear functions in the wanted matrices here, so that simple algorithms from linear algebra can be applied.

Normalized Radon Cumulative Distribution Transforms for Invariance and Robustness in Optimal Transport Based Image Classification

The Radon cumulative distribution transform (R-CDT), is an easy-to-compute feature extractor that facilitates image classification tasks especially in the small data regime. It is closely related to the sliced Wasserstein distance and provably guaranties the linear separability of image classes that emerge from translations or scalings. In many real-world applications, like the recognition of watermarks in filigranology, however, the data is subject to general affine transformations originating from the measurement process. To overcome this issue, we recently introduced the so-called max-normalized R-CDT that only requires elementary operations and guaranties the separability under arbitrary affine transformations. The aim of this paper is to continue our study of the max-normalized R-CDT especially with respect to its robustness against non-affine image deformations. Our sensitivity analysis shows that its separability properties are stable provided the Wasserstein-infinity distance between the samples can be controlled. Since the Wasserstein-infinity distance only allows small local image deformations, we moreover introduce a mean-normalized version of the R-CDT. In this case, robustness relates to the Wasserstein-2 distance and also covers image deformations caused by impulsive noise for instance. Our theoretical results are supported by numerical experiments showing the effectiveness of our novel feature extractors as well as their robustness against local non-affine deformations and impulsive noise.