Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.22826

Denoising Multi-Color QR Codes and Stiefel-Valued Data by Relaxed Regularizations

Robert Beinert·Jonas Bresch
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.OCcs.CVcs.NAmath.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robert Beinert, Jonas Bresch. 2025-06-28. Denoising Multi-Color QR Codes and Stiefel-Valued Data by Relaxed Regularizations. https://arxiv.org/abs/2506.22826

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the stability of proximal operators in Wasserstein spaces under different notions of convexity

The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.

math.OC↗
arXiv2026

Symmetry-dependence in Rounding of a Convex Body

The symmetry measure of a convex body $S\subset\mathbb{R}^n$ is given by: $\mathrm{sym}(S):=\max\{α\ge0:\text{ there exists }x\in S\text{ such that }-α(S-x)\subseteq S-x\}$, where such an $x$ is called a Minkowski center. We prove that every convex body $S$ admits a $\sqrt{\frac{n}{\mathrm{sym}(S)}}$-rounding of $S$, namely, there exists an origin-centered ellipsoid $E$ and a center $c$ such that $E\subseteq S-c\subseteq\sqrt{\frac{n}{\mathrm{sym}(S)}}\,E$. This result was conjectured in 2005 by Belloni and Freund. As special cases, this recovers an $n$-rounding of $S$ (since $\mathrm{sym}(S)\ge\frac{1}{n}$), and a $\sqrt{n}$-rounding when $\mathrm{sym}(S)=1$. In the case when $S$ is a polytope given as the convex hull of points, the desired rounding is produced by a regularized minimum-volume covering ellipsoid problem where the regularization is with respect to the Minkowski center. Similarly, when $S$ is a polytope given as the intersection of halfspaces, such a rounding is produced by a regularized maximum-volume inscribed ellipsoid problem. In both of these cases, the rounding can be computed by first solving a linear optimization problem (to compute $\mathrm{sym}(S)$ and a Minkowski center), and then solving a convex optimization problem with a logarithmic determinant objective, second-order cone constraints, and one semidefinite cone constraint. We also show that the factor $\sqrt{\frac{n}{\mathrm{sym}(S)}}$ is nearly tight in its dependence on dimension and symmetry. When $\frac{n+1}{1+\mathrm{sym}(S)}$ is an integer, we show by explicit construction that the factor $\sqrt{\frac{n}{\mathrm{sym}(S)}}$ is tight. In the more general case, for every dimension $n$ and every admissible symmetry value, we construct a polytope $S$ for which every rounding factor is at least $\sqrt{\frac{2}{3}}\sqrt{\frac{n}{\mathrm{sym}(S)}}$.

math.OC↗
arXiv2026

Improving the Last-Iterate Guarantees of Anytime Algorithms for Stochastic Monotone Variational Inequalities

We analyze a stochastic algorithm with Halpern-type anchoring for constrained convex-concave problems and monotone variational inequalities. This single-loop and single-call algorithm uses one unbiased sample of the gradient operator at every iteration, to be applicable to monotone games with noisy feedback. With $t$ denoting the iteration counter, we prove an anytime last-iterate convergence rate of $O(t^{-1/4})$ for both the gradient-mapping norm and restricted gap, bypassing the $O(t^{-1/5})$ constrained-anytime bottleneck in the literature. Specializing then to multi-point oracles, we use variance reduction to achieve the $O(t^{-1/2})$ rate with an anytime single-loop algorithm using $2$ samples per iteration. Our results allow constrained problems with a potentially unbounded feasible set; as well as a structured class of stochastic oracles whose variance need not be uniformly bounded.

math.OC↗