含噪非线性信息与熵数
Noisy nonlinear information and entropy numbers
AI总结：
本文用熵数刻画受确定性噪声干扰的最优（非）连续信息的质量，证明噪声下连续测量相对线性测量的增益有限但某些情况下显著。
摘要：
不可能用少于m次线性测量从R^m中恢复一个向量，即使测量是自适应选择的。最近研究表明，仅用O(log m)次连续（甚至Lipschitz）自适应测量即可任意精度地从R^m恢复向量，这使得连续信息相对于线性信息在各种逼近问题中获得了指数级加速。本文中，我们用熵数刻画受确定性噪声干扰的最优（非）连续信息的质量。这表明在存在噪声的情况下，连续测量相对于线性测量的潜在增益是有限的，但在某些情况下是显著的。
英文摘要：
It is impossible to recover a vector from $\mathbb{R}^m$ with less than $m$ linear measurements, even if the measurements are chosen adaptively. Recently, it has been shown that one can recover vectors from $\mathbb{R}^m$ with arbitrary precision using only $O(\log m)$ continuous (even Lipschitz) adaptive measurements, resulting in an exponential speed-up of continuous information compared to linear information for various approximation problems. In this note, we characterize the quality of optimal (dis-)continuous information that is disturbed by deterministic noise in terms of entropy numbers. This shows that in the presence of noise the potential gain of continuous over linear measurements is limited, but significant in some cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
