arXiv subjects

Robert Beinert

Publications and source records attributed to Robert Beinert.

At least 37 records · Page 2Linked to original sources

arXiv2023

Prony-Based Super-Resolution Phase Retrieval of Sparse, Multivariate Signals

Phase retrieval consists in the recovery of an unknown signal from phaseless measurements of its usually complex-valued Fourier transform. Without further assumptions, this problem is notorious to be severe ill posed such that the recovery of the true signal is nearly impossible. In certain applications like crystallography, speckle imaging in astronomy, or blind channel estimation in communications, the unknown signal has a specific, sparse structure. In this paper, we exploit these sparse structure to recover the unknown signal uniquely up to inevitable ambiguities as global phase shifts, transitions, and conjugated reflections. Although using a constructive proof essentially based on Prony's method, our focus lies on the derivation of a recovery guarantee for multivariate signals using an adaptive sampling scheme. Instead of sampling the entire multivariate Fourier intensity, we only employ Fourier samples along certain adaptively chosen lines. For bivariate signals, an analogous result can be established for samples in generic directions. The number of samples here scales quadratically to the sparsity level of the unknown signal.

cs.IT↗

arXiv2023

Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line

This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$ with the non-smooth negative distance kernel $K(x,y) = -|x-y|$, we deduce a formula for the associated functional. This functional appears to be convex, and we show that $\mathcal F_ν$ is convex along (generalized) geodesics. For the Dirac measure $ν= δ_q$, $q \in \mathbb R$ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration.

math.OC↗

arXiv2022

Multi-marginal Approximation of the Linear Gromov-Wasserstein Distance

Recently, two concepts from optimal transport theory have successfully been brought to the Gromov--Wasserstein (GW) setting. This introduces a linear version of the GW distance and multi-marginal GW transport. The former can reduce the computational complexity when computing all GW distances of a large set of inputs. The latter allows for a simultaneous matching of more than two marginals, which can for example be used to compute GW barycenters. The aim of this paper is to show an approximation result which characterizes the linear version as a limit of a multi-marginal GW formulation.

math.NA↗

arXiv2022

Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

We consider the imaging problem of the reconstruction of a three-dimensional object via optical diffraction tomography under the assumptions of the Born approximation. Our focus lies in the situation that a rigid object performs an irregular, time-dependent rotation under acoustical or optical forces. In this study, we compare reconstruction algorithms in case i) that two-dimensional images of the complex-valued wave are known, or ii) that only the intensity (absolute value) of these images can be measured, which is the case in many practical setups. The latter phase-retrieval problem can be solved by an all-at-once approach based utilizing a hybrid input-output scheme with TV regularization.

math.NA↗

arXiv2022

On a linear Gromov-Wasserstein distance

Gromov-Wasserstein distances are generalization of Wasserstein distances, which are invariant under distance preserving transformations. Although a simplified version of optimal transport in Wasserstein spaces, called linear optimal transport (LOT), was successfully used in practice, there does not exist a notion of linear Gromov-Wasserstein distances so far. In this paper, we propose a definition of linear Gromov-Wasserstein distances. We motivate our approach by a generalized LOT model, which is based on barycentric projection maps of transport plans. Numerical examples illustrate that the linear Gromov-Wasserstein distances, similarly as LOT, can replace the expensive computation of pairwise Gromov-Wasserstein distances in applications like shape classification.

math.NA↗

arXiv2022

Total variation-based reconstruction and phase retrieval for diffraction tomography

In optical diffraction tomography (ODT), the three-dimensional scattering potential of a microscopic object rotating around its center is recovered by a series of illuminations with coherent light. Reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane, whereas often only intensities, i.e., phaseless measurements, are available in practice. We propose a new reconstruction approach for ODT with unknown phase information based on three key ingredients. First, the light propagation is modeled using Born's approximation enabling us to use the Fourier diffraction theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier transform via total variation regularization utilizing a primal-dual iteration, which also yields a novel numerical inversion formula for ODT with known phase. The third ingredient is a hybrid input-output scheme. We achieved convincing numerical results, which indicate that ODT with phaseless data is possible. The so-obtained 2D and 3D reconstructions are even comparable to the ones with known phase.

math.NA↗

arXiv2022

On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line

Let $x_1 < \dots < x_n$ and $y_1 < \dots < y_n$, $n \in \mathbb N$, be real numbers. We show by an example that the assignment problem $$ \max_{σ\in S_n} F_σ(x,y) := \frac12 \sum_{i,k=1}^n |x_i - x_k|^α\, |y_{σ(i)} - y_{σ(k)}|^α, \quad α>0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n > 2 +2^α$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\text{id}(x,y)$ and $F_\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.

math.NA↗

arXiv2021

Approximation of Curve-based Sleeve Functions in High Dimensions

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function - the profile - is recovered, and second the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximate the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

math.NA↗

arXiv2021

On the Dynamical System of Principal Curves in $\mathbb R^d$

Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based on the conditional expectation and from the variational-calculus point of view as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Duchamp and Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to $\mathbb R^d$ with $d \ge 3$. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in $\mathbb R^3$.

math.DS↗

arXiv2021

Phase Retrieval and System Identification in Dynamical Sampling via Prony's Method

Phase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e. a sequence of sampling vectors constructed by the repeated action of an operator. The loss of the phase here turns the well-posed dynamical sampling into a severe ill-posed inverse problem. In the existing literature, the involved operator is usually completely known. In this paper, we combine phase retrieval in dynamical sampling with the identification of the system. For instance, if the dynamical frame is based on a repeated convolution, then we want to recover the unknown convolution kernel in advance. Using Prony's method, we establish several recovery guarantees for signal and system, whose proofs are constructive and yield analytic recovery methods. The required assumptions are satisfied by almost all signals, operators, and sampling vectors. Moreover, these guarantees not only hold for the finite-dimensional setting but also carry over to infinite-dimensional spaces. Studying the sensitivity of the analytic recovery procedures, we also establish error bounds for the applied approximate Prony method with respect to complex exponential sums.

math.NA↗

arXiv2021

Phase Retrieval via Polarization in Dynamical Sampling

In this paper we consider the nonlinear inverse problem of phase retrieval in the context of dynamical sampling. Where phase retrieval deals with the recovery of signals & images from phaseless measurements, dynamical sampling was introduced by Aldroubi et al in 2015 as a tool to recover diffusion fields from spatiotemporal samples. Considering finite-dimensional signals evolving in time under the action of a known matrix, our aim is to recover the signal up to global phase in a stable way from the absolute value of certain space-time measurements. First, we state necessary conditions for the dynamical system of sampling vectors to make the recovery of the unknown signal possible. The conditions deal with the spectrum of the given matrix and the initial sampling vector. Then, assuming that we have access to a specific set of further measurements related to aligned sampling vectors, we provide a feasible procedure to recover almost every signal up to global phase using polarization techniques. Moreover, we show that by adding extra conditions like full spark, the recovery of all signals is possible without exceptions.

math.NA↗

arXiv2021

Super-Resolution for Doubly-Dispersive Channel Estimation

In this work we consider the problem of identification and reconstruction of doubly-dispersive channel operators which are given by finite linear combinations of time-frequency shifts. Such operators arise as time-varying linear systems for example in radar and wireless communications. In particular, for information transmission in highly non-stationary environments the channel needs to be estimated quickly with identification signals of short duration and for vehicular application simultaneous high-resolution radar is desired as well. We consider the time-continuous setting and prove an exact resampling reformulation of the involved channel operator when applied to a trigonometric polynomial as identifier in terms of sparse linear combinations of real-valued atoms. Motivated by recent works of Heckel et al. we present an exact approach for off-the-grid superresolution which allows to perform the identification with realizable signals having compact support. Then we show how an alternating descent conditional gradient algorithm can be adapted to solve the reformulated problem. Numerical examples demonstrate the performance of this algorithm, in particular in comparison with a simple adaptive grid refinement strategy and an orthogonal matching pursuit algorithm.

cs.IT↗

arXiv2020

Robust PCA via Regularized REAPER with a Matrix-Free Proximal Algorithm

Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called REAPER, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must be determined in advance and the minimization is performed over symmetric positive semi-definite matrices having the size of the data, although the number of principal components is substantially smaller. This prohibits its use if the dimension of the data is large which is often the case in image processing. In this paper, we propose a regularized version of REAPER which enforces the sparsity of the number of principal components by penalizing the nuclear norm of the corresponding orthogonal projector. This has the advantage that only an upper bound on the number of principal components is required. Our second contribution is a matrix-free algorithm to find a minimizer of the regularized REAPER which is also suited for high dimensional data. The algorithm couples a primal-dual minimization approach with a thick-restarted Lanczos process. As a side result, we discuss the topic of the bias in robust PCA. Numerical examples demonstrate the performance of our algorithm.

math.NA↗

arXiv2019

Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval

We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, a direct realization of these algorithms for, e.g., image recovery problems is often impracticable, since computations have to be performed on the tensor-product space, whose dimension is usually tremendous. To overcome this limitation, we derive tensor-free versions of common singular value thresholding methods by exploiting low-rank representations and incorporating an augmented Lanczos process. Using a novel reweighting technique, we further improve the convergence behavior and rank evolution of the iterative algorithms. Applying the method to the two-dimensional masked Fourier phase retrieval problem, we obtain an efficient recovery method. Moreover, the tensor-free algorithms are flexible enough to incorporate a-priori smoothness constraints that greatly improve the recovery results.

math.OC↗

arXiv2018

Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting

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.

math.NA↗

arXiv2017

Fourier Phase Retrieval: Uniqueness and Algorithms

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field.

cs.IT↗

arXiv2017

Sparse phase retrieval of one-dimensional signals by Prony's method

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.

math.NA↗

arXiv2016

Ambiguities in one-dimensional phase retrieval from magnitudes of a linear canonical transform

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.

math.NA↗