Stability in unlimited sampling

2026-03-25

AI Summary

This work addresses the inherent instability in folded sampling for reconstructing bandlimited signals, which arises from sampling saturation. For the first time, the problem is formulated as a shortest vector problem on an infinite-dimensional lattice, and signal energy constraints are introduced to restore stability. By leveraging the spectral properties of Fourier concentration matrices together with bounds from integer Chebyshev polynomials, the authors establish a unified theoretical framework that rigorously proves energy-constrained reconstruction remains stable under both uniform and non-uniform sampling. This framework not only guarantees stable recovery but also provides a coherent theoretical foundation that unifies recent results on injectivity and encoding guarantees, thereby laying the groundwork for practical applications of folded sampling.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Folded sampling replaces clipping in analog-to-digital converters by reducing samples modulo a threshold, thereby avoiding saturation artifacts. We study the reconstruction of bandlimited functions from folded samples and show that, for equispaced sampling patterns, the recovery problem is inherently unstable. We then prove that imposing any a priori energy bound restores stability, and that this regularization effect extends to non-uniform sampling geometries. Our analysis recasts folded-sampling stability as an infinite-dimensional lattice shortest-vector problem, which we resolve via harmonic-analytic tools (the spectral profile of Fourier concentration matrices) and, alternatively, via bounds for integer Tschebyschev polynomials. Our work brings context to recent results on injectivity and encoding guarantees for folded sampling and further supports the empirical success of folded sampling under natural energy constraints.

Problem

• folded sampling
• stability
• bandlimited functions
• reconstruction
• sampling

Innovation

• folded sampling
• stability
• energy constraint
• Fourier concentration
• integer Chebyshev polynomials