Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34842

A Variational Geometric Approach to Exact Recovery by $\ell_1/\ell_2$ Minimization

Yulan Liu·Rongrong Lin
View original source ↗Read source document ↗Export metadata ↓

Abstract

For affine-constrained $\ell_1/\ell_2$ minimization with nonzero data, a variational geometric approach is developed to characterize exact recovery. Necessary and sufficient conditions for local and global exact recovery are established in terms of directional derivatives and cones associated with sublevel sets of the ratio objective. The local analysis further reveals that local minimality, first-order growth, and tilt stability are equivalent. These pointwise characterizations lead to improved sufficient conditions for uniform recovery of sparse signals. The sufficient condition for uniform local recovery has coefficient $c_s=\max\{1,(\sqrt{s}-1)/2\}$ for $s\ge2$. For each fixed $s$, the coefficient $c_s$ is optimal within the considered family of local null-space conditions. Projection estimates yield a scalar bound for uniform global recovery with coefficient two, improving the earlier coefficient three. The coefficient two is optimal within the considered family of global null-space bounds when both $s$ and $n$ may vary. A positive margin in the local null-space condition provides explicit growth moduli, localization radii, and bounds on linear objective perturbations that leave the localized minimizer fixed. Examples show that uniform local recovery does not imply uniform global recovery.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yulan Liu, Rongrong Lin. 2026-10-26. A Variational Geometric Approach to Exact Recovery by $\ell_1/\ell_2$ Minimization. https://arxiv.org/abs/2610.34842

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

KEEP EXPLORING

Related papers

arXiv2026

The Robust Newton Method for Polynomials and its Polynomiography

We present the Robust Newton Method (RNM) for a complex polynomial $p$ of degree $n\geq2$, with continuous map $F(z)=z-p(z)\overline{p'(z)}/(9A(z)^2)$, where $A(z)=\max_{0\leq j\leq n}|p^{(j)}(z)|/j!$. Its fixed points are the roots and critical points; elsewhere it strictly decreases $|p|$ along the Newton direction. The Geometric Modulus Principle inspires the construction and supplies an escape step at nonroot critical points. Every root has an open basin. When nonroot critical points are simple, almost every seed converges to a root, and root-basin boundaries lie in countably many Lipschitz curves with Hausdorff dimension one when nonempty. These conclusions apply after any sufficiently small nonzero linear perturbation $p(z)+δz$. Every RNM orbit, with or without escapes, converges to a root or critical point. A hybrid enforcing one RNM update every $T$ iterations retains convergence and reaches $|p(z)p'(z)|\leq\varepsilon$ in $O(T/\varepsilon^2)$ updates, with constants depending on the polynomial and seed; $T=1$ gives RNM. Orbits approaching simple roots eventually reach Smale's $α$-certification, permitting a permanent Newton switch and $O(\log\log(1/\varepsilon))$ further steps for absolute root error $\varepsilon$. For cubics, deterministic critical-point initialization yields an explicit preparatory bound, retained by an RNM-guided Newton refinement. Illustrative computations cover cubics and degree-ten polynomials. For $p(z)=z^n-1$, the basins are exactly the open Voronoi regions, with boundary points converging to the origin. Polynomiographs illustrate basin geometry and iteration-based coloring.

math.NA↗
arXiv2026

High-performance computing for the BGK model of the Boltzmann equation with a meshfree Arbitrary Lagrangian-Eulerian method

In this paper, we present high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method. For the numerical simulation of the BGK equation, we use an Arbitrary-Lagrangian-Eulerian (ALE) method developed in previous work, where the approximation of spatial derivatives and the reconstruction of a function are based on the weighted least squares method. A Graphics Processing Unit (GPU) is used to accelerate the code. The result is compared with sequential and parallel versions of the CPU code. Two- and three-dimensional driven cavity problems are solved, where a speed-up of several orders of magnitude is obtained compared to a sequential CPU simulation. The transport step is implemented as a node-parallel CUDA kernel for both two- and three-dimensional physical space: one GPU thread advances one particle-velocity-node pair by tracing its characteristic foot point and reconstructing the distribution there with a register-resident weighted least squares solve. For the three-dimensional case, no velocity reduction is applicable: a single distribution function $f$ is evolved on the full $N_v^3$ velocity grid, so the transport in every coordinate direction follows the resolved velocity node. The device memory footprint is kept at exactly twice the distribution storage by a structure-of-arrays layout with a single transport buffer; particle management runs on the host, so no device-side scratch copies of the cloud are required.

math.NA↗
arXiv2026

A Computational Method for the Inverse Robin Problem with Convergence Rate

The inverse Robin problem consists in determining the Robin parameter in an elliptic partial differential equation posed on a domain $Ω$. In this work it is assumed that the right-hand sides of the elliptic problem are known, that the associated solution is available on a subdomain $ω\subset Ω$ and that this solution vanishes at most at isolated points on the boundary $\partial Ω$. Within this framework, a computational method for the reconstruction of the Robin parameter is established, which is inspired by a unique continuation method. The proposed scheme relies solely on first-order Lagrange finite elements ensuring a straightforward implementation. Under the main assumption that the Robin parameter is in a finite dimensional space of continuously differentiable functions it is shown that the numerical method is second order convergent in the finite element's mesh size. For noisy data this convergence rate is shown to hold true until the noise term dominates the error estimate. Numerical experiments are presented that highlight the feasibility of the Robin parameter reconstruction and that confirm the theoretical convergence results numerically.

math.NA↗