Noisy nonlinear information and entropy numbers
David Krieg, Erich Novak, Leszek Plaskota, Mario Ullrich
TL;DR
This work links the quality of approximations under noisy information to entropy numbers, clarifying how much advantage adaptive or continuous information can provide. By framing the problem in information-based complexity, it shows that for arbitrary measurements the minimal error $e_n^{\mathrm{arb}}(S,\delta)$ is tightly bounded by entropy numbers $\varepsilon_n(S)$ through explicit factors depending on the noise level $\delta$, while for continuous measurements $e_n^{\mathrm{con}}(S,\delta)$ is always bounded above by $\varepsilon_n(S)$ and, in the large-noise regime, bounded below by $\varepsilon_{b n}(S)$. The paper also demonstrates that continuous information can exhibit significant gains in certain regimes (notably when singular values decay rapidly or in finite-dimensional settings) but that, in general with noise, the potential advantage over linear information is limited; the diagonal-operator analysis provides concrete formulas and asymptotics illustrating these limits and abilities. Overall, the results give sharp, entropy-number–driven insights into using nonlinear and adaptive information under deterministic noise, with practical implications for high-dimensional approximation and operator compression.
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.