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.

翻译：从有限测量值重构无限维信号是逼近论与信号处理中的基本问题。虽然广义采样（GS）框架为任意可分离希尔伯特空间中的元素恢复提供了稳健方法论，但确定性方法受限于严重的基依赖维度约束，通常需要二次样本复杂度 $m \gtrsim n^2$ 以避免数值不稳定性。本文提出了一种完全随机的GS框架，从根本上突破了这些确定性障碍。通过根据最优杠杆得分概率分布进行测量，我们证明在接近线性的样本复杂度 $m \gtrsim n\log n$ 下，稳定恢复可高概率实现。关键的是，这一最优速率具有普适性——独立于测量基与重构基的具体选择，甚至适用于高度冗余框架的感知系统。为建立这些保证，我们推导了随机矩形算子的新矩阵伯恩斯坦不等式，从而能够严格受控于经验交叉项主导的混叠误差。最后，我们在通过勒让德多项式从连续傅里叶测量中恢复解析函数的经典问题上验证了该方法实际效能，所提随机化方法可实现近指数收敛速率。