Stochastic Generalized Sampling

2026-05-22

AI Summary

This work addresses the quadratic sample complexity and numerical instability inherent in conventional generalized sampling for infinite-dimensional signal reconstruction, which arise from dependence on specific bases. To overcome these limitations, the authors propose a fully randomized generalized sampling framework that draws samples according to an optimal leverage score distribution. By leveraging a novel matrix Bernstein inequality for rectangular random operators and enforcing rigorous aliasing error control, the method transcends the dimensional constraints of deterministic approaches. The resulting framework achieves near-linear sample complexity independent of both the measurement and reconstruction bases, making it suitable for highly redundant systems. Notably, when applied to continuous Fourier measurements for recovering analytic functions via Legendre polynomials, the approach attains near-exponential convergence rates, substantially enhancing reconstruction efficiency and numerical stability.

Abstract

Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for recovering elements in arbitrary separable Hilbert spaces, deterministic approaches suffer from severe basis-dependent dimensionality constraints, often requiring a quadratic sample complexity $m \gtrsim n^2$ to avoid numerical instability. In this paper, we introduce a fully stochastic framework for GS that natively overcomes these deterministic barriers. By drawing measurements according to an optimal leverage-score probability distribution, we prove that stable recovery is guaranteed with high probability at a near-linear sample complexity of $m \gtrsim n\log n$. Crucially, this optimal rate is universal-independent of the specific choice of measurement and reconstruction bases-and holds even when the sensing system is a highly redundant frame. To establish these guarantees, we derive a novel matrix Bernstein inequality for random rectangular operators, allowing us to rigorously control the aliasing error governed by the empirical cross-term. Finally, we demonstrate the practical efficacy of our approach on the classical problem of recovering analytic functions from continuous Fourier measurements via Legendre polynomials, where our randomized method achieve near-exponential convergence rates.

Problem

• Generalized Sampling
• Sample Complexity
• Infinite-dimensional Signal Reconstruction
• Basis-dependent Constraints
• Stable Recovery

Innovation

• stochastic generalized sampling
• leverage scores
• near-linear sample complexity
• matrix Bernstein inequality
• aliasing error