Robert Beinert

Robert Beinert, Kristian Bredies

Considering the question: how non-linear may a non-linear operator be in order to extend the linear regularization theory, we introduce the class of dilinear mappings, which covers linear, bilinear, and quadratic operators between Banach spaces. The corresponding dilinear inverse problems cover blind deconvolution, deautoconvolution, parallel imaging in MRI, and the phase retrieval problem. Based on the universal property of the tensor product, the central idea is here to lift the non-linear mappings to linear representatives on a suitable topological tensor space. At the same time, we extend the class of usually convex regularization functionals to the class of diconvex functionals, which are likewise defined by a tensorial lifting. Generalizing the concepts of subgradients and Bregman distances from convex analysis to the new framework, we analyse the novel class of dilinear inverse problems and establish convergence rates under similar conditions than in the linear setting. Considering the deautoconvolution problem as specific application, we derive satisfiable source conditions and validate the theoretical convergence rates numerically.

Robert Beinert, Gerlind Plonka

Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.

Robert Beinert, Gerlind Plonka

In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from O(N^2) Fourier intensity measurements up to trivial ambiguities. The constructive proof consists of two steps, where in the first step the Prony method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.

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

Phase retrieval problems occur in a width range of applications in physics and engineering such as crystallography, astronomy, and laser optics. Common to all of them is the recovery of an unknown signal from the intensity of its Fourier transform. Because of the well-known ambiguousness of these problems, the determination of the original signal is generally challenging. Although there are many approaches in the literature to incorporate the assumption of non-negativity of the solution into numerical algorithms, theoretical considerations about the solvability with this constraint occur rarely. In this paper, we consider the one-dimensional discrete-time setting and investigate whether the usually applied a priori non-negativity can overcame the ambiguousness of the phase retrieval problem or not. We show that the assumed non-negativity of the solution is usually not a sufficient a priori condition to ensure uniqueness in one-dimensional phase retrieval. More precisely, using an appropriate characterization of the occurring ambiguities, we show that neither the uniqueness nor the ambiguousness are rare exceptions.

Robert Beinert

The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumption or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference measurements with a known or an unknown reference which is unrelated to the unknown signal.

Robert Beinert

Phase retrieval problems occur in a wide range of applications in physics and engineering. Usually, these problems consist in the recovery of an unknown signal from the magnitudes of its Fourier transform. In some applications, however, the given intensity arises from a different transformation such as the Fresnel or fractional Fourier transform. More generally, we here consider the phase retrieval of an unknown signal from the magnitudes of an arbitrary linear canonical transform. Using the close relation between the Fourier and the linear canonical transform, we investigate the arising ambiguities of these phase retrieval problems and transfer the well-known characterizations of the solution sets from the classical Fourier phase retrieval problem to the new setting.

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) 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.

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.

Moritz Piening, Robert Beinert

Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space $L_2([0,1])$. For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional $L_2$-projections, parametrized by Gaussian processes. Combining this 1d construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.

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.