Noisy nonlinear information and entropy numbers

TL;DR

This paper investigates the limits of recovering vectors from noisy nonlinear measurements, showing that noise constrains the advantage of continuous measurements over linear ones, with implications for information theory and approximation.

Contribution

It characterizes the effectiveness of optimal noisy continuous and discontinuous measurements using entropy numbers, highlighting the impact of noise on measurement efficiency.

Findings

Noise limits the advantage of continuous measurements over linear ones

Optimal measurement quality can be characterized by entropy numbers

Continuous measurements still offer significant benefits in certain noisy scenarios

Abstract

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.