Robert Beinert

Paul Hagemann, Johannes Hertrich, Fabian Altekrüger et al.

We propose conditional flows of the maximum mean discrepancy (MMD) with the negative distance kernel for posterior sampling and conditional generative modeling. This MMD, which is also known as energy distance, has several advantageous properties like efficient computation via slicing and sorting. We approximate the joint distribution of the ground truth and the observations using discrete Wasserstein gradient flows and establish an error bound for the posterior distributions. Further, we prove that our particle flow is indeed a Wasserstein gradient flow of an appropriate functional. The power of our method is demonstrated by numerical examples including conditional image generation and inverse problems like superresolution, inpainting and computed tomography in low-dose and limited-angle settings.

Robert Beinert, Jonas Bresch, Michael Quellmalz

The Radon transform is a fundamental tool for analyzing data in tomographic imaging, optimal transport, crystallography, and geometric analysis. Numerical computations require an accurate discretization. To deal with voxelized images and objects, we derive a closed-form, piecewise polynomial expression for the Radon transform of an axis-aligned cube in arbitrary dimension $d$. Building on this formula, we propose a discrete Radon transform in $\mathbb{R}^d$ that is both analytically exact for voxelized data and computationally efficient. For improved numerical stability, we introduce a regularized variant replacing the Radon transform of a cube, i.e.\ the $(d-1)$-dimensional area of the intersection between that cube and a hyperplane, by the $d$-dimensional volume of the intersection between the cube and a thin slab around the hyperplane. Numerical experiments demonstrate the effectiveness of the proposed approach in several applications including 3D shape matching, classification, and sliced Wasserstein barycenters. The computational efficiency in higher dimensions is verified by a comparison with Monte Carlo integration.

Matthias Beckmann, Robert Beinert, Jonas Bresch

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.

Florian Beier, Moritz Piening, Robert Beinert et al.

We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.

Moritz Piening, Robert Beinert

Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modeling. A key challenge in working with GMMs is defining a computationally efficient and geometrically meaningful metric. The mixture Wasserstein (MW) distance adapts the Wasserstein metric to GMMs and has been applied in various domains, including domain adaptation, dataset comparison, and reinforcement learning. However, its high computational cost -- arising from repeated Wasserstein distance computations involving matrix square root estimations and an expensive linear program -- limits its scalability to high-dimensional and large-scale problems. To address this, we propose multiple novel slicing-based approximations to the MW distance that significantly reduce computational complexity while preserving key optimal transport properties. From a theoretical viewpoint, we establish several weak and strong equivalences between the introduced metrics, and show the relations to the original MW distance and the well-established sliced Wasserstein distance. Furthermore, we validate the effectiveness of our approach through numerical experiments, demonstrating computational efficiency and applications in clustering, perceptual image comparison, and GMM minimization

Moritz Piening, Robert Beinert

The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.

Robert Beinert, Jonas Bresch

The handling of manifold-valued data, for instance, plays a central role in color restoration tasks relying on circle- or sphere-valued color models, in the study of rotational or directional information related to the special orthogonal group, and in Gaussian image processing, where the pixel statistics are interpreted as values on the hyperbolic sheet. Especially, to denoise these kind of data, there have been proposed several generalizations of total variation (TV) and Tikhonov-type denoising models incorporating the underlying manifolds. Recently, a novel, numerically efficient denoising approach has been introduced, where the data are embedded in an Euclidean ambient space, the non-convex manifolds are encoded by a series of positive semi-definite, fixed-rank matrices, and the rank constraint is relaxed to obtain a convexification that can be solved using standard algorithms from convex analysis. The aim of the present paper is to extent this approach to new kinds of data like multi-binary and Stiefel-valued data. Multi-binary data can, for instance, be used to model multi-color QR codes whereas Stiefel-valued data occur in image and video-based recognition. For both new data types, we propose TV- and Tikhonov-based denoising modelstogether with easy-to-solve convexification. All derived methods are evaluated on proof-of-concept, synthetic experiments.