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.

翻译：从少于$m$个线性测量中恢复$\mathbb{R}^m$中的向量是不可能的，即使测量是自适应选择的。最近的研究表明，仅使用$O(\log m)$个连续（甚至Lipschitz）自适应测量即可以任意精度恢复$\mathbb{R}^m$中的向量，这使得在多种逼近问题中连续信息相较于线性信息实现了指数级加速。本文通过熵数刻画了受确定性噪声干扰的最优（非）连续信息的质量。结果表明，在噪声存在的情况下，连续测量相对于线性测量的潜在增益是有限的，但在某些情况下仍具有显著意义。