arXiv subjects

Kristian Bredies

Publications and source records attributed to Kristian Bredies.

At least 19 recordsLinked to original sources

arXiv2025

Learning Firmly Nonexpansive Operators

This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock primal-dual iteration, Douglas-Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate way. Finally, the experimental section details practical implementations of the method and presents an application in image denoising, where we consider a novel, interpretable PnP Chambolle-Pock primal-dual iteration.

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arXiv2025

On extremal points for some vectorial total variation seminorms

We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and sparse optimization algorithms for variational problems posed among such functions. For cases in which either the domain or the target are one dimensional or the sum of the total variations of each component is used, we prove that these extremals consist of piecewise constant functions with two regions. For definitions involving more involved matrix norms and in particular spectral norms, we produce families of examples to show that the resulting set of extremal points is larger and includes piecewise constant functions with more than two regions. We also consider the total deformation induced by the symmetrized gradient, for which minimization with linear constraints appears in problems of determination of limit loads in a number of continuum mechanical models involving plasticity. For this case, we show piecewise infinitesimally rigid functions with two pieces to be extremal under mild assumptions. Finally, as an example which is not piecewise constant, we prove that unit radial vector fields are extremal for the Frobenius total variation in the plane.

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arXiv2025

A Second-Order TGV Discretization with $90^{\circ}$ Rotational Invariance Property

In this work, we propose a new discretization for second-order total generalized variation (TGV) with some distinct properties compared to existing discrete formulations. The introduced model is based on same design principles as Condat's discrete total variation model (\textit{SIAM J. Imaging Sci}., 10(3), 1258--1290, 2017) and shares its benefits, in particular, improved quality for the solution of imaging problems. An algorithm for image denoising with second-order TGV using the new discretization is proposed. Numerical results obtained with this algorithm demonstrate the discretization's advantages. Moreover, in order to compare invariance properties of the new model, an algorithm for calculating the TGV value with respect to the new discretization model is given.

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arXiv2024

A sparse optimization approach to infinite infimal convolution regularization

In this paper we introduce the class of infinite infimal convolution functionals and apply these functionals to the regularization of ill-posed inverse problems. The proposed regularization involves an infimal convolution of a continuously parametrized family of convex, positively one-homogeneous functionals defined on a common Banach space $X$. We show that, under mild assumptions, this functional admits an equivalent convex lifting in the space of measures with values in $X$. This reformulation allows us to prove well-posedness of a Tikhonov regularized inverse problem and opens the door to a sparse analysis of the solutions. In the case of finite-dimensional measurements we prove a representer theorem, showing that there exists a solution of the inverse problem that is sparse, in the sense that it can be represented as a linear combination of the extremal points of the ball of the lifted infinite infimal convolution functional. Then, we design a generalized conditional gradient method for computing solutions of the inverse problem without relying on an a priori discretization of the parameter space and of the Banach space $X$. The iterates are constructed as linear combinations of the extremal points of the lifted infinite infimal convolution functional. We prove a sublinear rate of convergence for our algorithm and apply it to denoising of signals and images using, as regularizer, infinite infimal convolutions of fractional-Laplacian-type operators with adaptive orders of smoothness and anisotropies.

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arXiv2024

A stochastic preconditioned Douglas-Rachford splitting method for saddle-point problems

In this article, we propose and study a stochastic and relaxed preconditioned Douglas--Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas--Rachford splitting methods.

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arXiv2023

Boundedness and unboundedness in total variation regularization

We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for sufficiently small noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. Finally, we discuss the possibility of extending such results to related higher-order regularization functionals, obtaining a positive answer for the infimal convolution of first and second order total variation.

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arXiv2023

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

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arXiv2023

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

We propose a fully-corrective generalized conditional gradient method (FC-GCG) for the minimization of the sum of a smooth, convex loss function and a convex one-homogeneous regularizer over a Banach space. The algorithm relies on the mutual update of a finite set $\mathcal{A}_k$ of extremal points of the unit ball of the regularizer and of an iterate $u_k \in \operatorname{cone}(\mathcal{A}_k)$. Each iteration requires the solution of one linear problem to update $\mathcal{A}_k$ and of one finite dimensional convex minimization problem to update the iterate. Under standard hypotheses on the minimization problem we show that the algorithm converges sublinearly to a solution. Subsequently, imposing additional assumptions on the associated dual variables, this is improved to a linear rate of convergence. The proof of both results relies on two key observations: First, we prove the equivalence of the considered problem to the minimization of a lifted functional over a particular space of Radon measures using Choquet's theorem. Second, the FC-GCG algorithm is connected to a Primal-Dual-Active-point Method (PDAP) on the lifted problem for which we finally derive the desired convergence rates.

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arXiv2022

Graph and distributed extensions of the Douglas-Rachford method

In this paper, we propose several graph-based extensions of the Douglas-Rachford splitting (DRS) method to solve monotone inclusion problems involving the sum of $N$ maximal monotone operators. Our construction is based on a two-layer architecture that we refer to as bilevel graphs, to which we associate a generalization of the DRS algorithm that presents the prescribed structure. The resulting schemes can be understood as unconditionally stable frugal resolvent splitting methods with a minimal lifting in the sense of Ryu [Math Program 182(1):233-273, 2020], as well as instances of the (degenerate) Preconditioned Proximal Point method, which provides robust convergence guarantees. We further describe how the graph-based extensions of the DRS method can be leveraged to design new fully distributed protocols. Applications to a congested optimal transport problem and to distributed Support Vector Machines show interesting connections with the underlying graph topology and highly competitive performances with state-of-the-art distributed optimization approaches.

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arXiv2022

A hybrid proximal generalized conditional gradient method and application to total variation parameter learning

In this paper we present a new method for solving optimization problems involving the sum of two proper, convex, lower semicontinuous functions, one of which has Lipschitz continuous gradient. The proposed method has a hybrid nature that combines the usual forward-backward and the generalized conditional gradient method. We establish a convergence rate of $o(k^{-1/3})$ under mild assumptions with a specific step-size rule and show an application to a total variation parameter learning problem, which demonstrates its benefits in the context of nonsmooth convex optimization.

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arXiv2022

A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

We study measure-valued solutions of the inhomogeneous continuity equation $\partial_t ρ_t + {\rm div}(vρ_t) = g ρ_t$ where the coefficients $v$ and $g$ are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger-Kantorovich energy is finite. This principle gives a decomposition of the solution into curves $t \mapsto h(t)δ_{γ(t)}$ that satisfy the characteristic system $\dot γ(t) = v(t, γ(t))$, $\dot h(t) = g(t, γ(t)) h(t)$ in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of $g$ where characteristics are not unique with respect to $h$. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger-Kantorovich-type regularizers are characterized. Such regularizers arise, e.g., in the context of dynamic inverse problems and dynamic optimal transport.

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arXiv2022

A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in (Bredies and Fanzon, ESAIM: M2AN, 54:2351-2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou-Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou-Brenier energy (Bredies et al., arXiv:1907.11589, 2019) we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists in iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimize the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model at reconstructing heavily undersampled dynamic data, together with the presence of noise.

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arXiv2021

Regularization Graphs -- A unified framework for variational regularization of inverse problems

We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear operators and convex functionals, assembled by means of operators that can be seen as generalizations of classical infimal convolution operators. This class of functionals exhaustively covers existing regularization approaches and it is flexible enough to craft new ones in a simple and constructive way. We provide well-posedness and convergence results with the proposed class of functionals in a general setting. Further, we consider a bilevel optimization approach to learn optimal weights for such regularization graphs from training data. We demonstrate that this approach is capable of optimizing the structure and the complexity of a regularization graph, allowing, for example, to automatically select a combination of regularizers that is optimal for given training data.

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arXiv2021

Degenerate Preconditioned Proximal Point algorithms

In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of splitting methods for monotone inclusion and convex minimization problems. Moreover, we show that the degeneracy of the preconditioner allows for a reduction of the variables involved in the iteration updates. We show the strength of the proposed framework in the context of splitting algorithms, providing new simplified proofs of convergence and highlighting the link between existing schemes, such as Chambolle-Pock, Forward Douglas-Rachford and Peaceman-Rachford, that we study from a preconditioned proximal point perspective. The proposed framework allows to devise new flexible schemes and provides new ways to generalize existing splitting schemes to the case of the sum of many terms. As an example, we present a new sequential generalization of Forward Douglas-Rachford along with numerical experiments that demonstrate its interest in the context of nonsmooth convex optimization.

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arXiv2021

Convergence analysis of pixel-driven Radon and fanbeam transforms

This paper presents a novel mathematical framework for understanding pixel-driven approaches for the parallel beam Radon transform as well as for the fanbeam transform, showing that with the correct discretization strategy, convergence - including rates - in the $L^2$ operator norm can be obtained. These rates inform about suitable strategies for discretization of the occurring domains/variables, and are first established for the Radon transform. In particular, discretizing the detector in the same magnitude as the image pixels (which is standard practice) might not be ideal and in fact, asymptotically smaller pixels than detectors lead to convergence. Possible adjustments to limited-angle and sparse-angle Radon transforms are discussed, and similar convergence results are shown. In the same vein, convergence results are readily extended to a novel pixel-driven approach to the fanbeam transform. Numerical aspects of the discretization scheme are discussed, and it is shown in particular that with the correct discretization strategy, the typical high-frequency artifacts can be avoided.

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arXiv2021

Non-linear fitting with joint spatial regularization in Arterial Spin Labeling

Multi-Delay single-shot arterial spin labeling (ASL) imaging provides accurate cerebral blood flow (CBF) and, in addition, arterial transit time (ATT) maps but the inherent low SNR can be challenging. Especially standard fitting using non-linear least squares often fails in regions with poor SNR, resulting in noisy estimates of the quantitative maps. State-of-the-art fitting techniques improve the SNR by incorporating prior knowledge in the estimation process which typically leads to spatial blurring. To this end, we propose a new estimation method with a joint spatial total generalized variation regularization on CBF and ATT. This joint regularization approach utilizes shared spatial features across maps to enhance sharpness and simultaneously improves noise suppression in the final estimates. The proposed method is evaluated at three levels, first on synthetic phantom data including pathologies, followed by in vivo acquisitions of healthy volunteers, and finally on patient data following an ischemic stroke. The quantitative estimates are compared to two reference methods, non-linear least squares fitting and a state-of-the-art ASL quantification algorithm based on Bayesian inference. The proposed joint regularization approach outperforms the reference implementations, substantially increasing the SNR in CBF and ATT while maintaining sharpness and quantitative accuracy in the estimates.

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arXiv2020

On the extremal points of the ball of the Benamou-Brenier energy

In this paper we characterize the extremal points of the unit ball of the Benamou--Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite dimensional data and optimal-transport based regularization.

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arXiv2020

Inverse problems with second-order Total Generalized Variation constraints

Total Generalized Variation (TGV) has recently been introduced as penalty functional for modelling images with edges as well as smooth variations. It can be interpreted as a "sparse" penalization of optimal balancing from the first up to the $k$-th distributional derivative and leads to desirable results when applied to image denoising, i.e., $L^2$-fitting with TGV penalty. The present paper studies TGV of second order in the context of solving ill-posed linear inverse problems. Existence and stability for solutions of Tikhonov-functional minimization with respect to the data is shown and applied to the problem of recovering an image from blurred and noisy data.

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