How many continuous measurements are needed to learn a vector?

TL;DR

This paper demonstrates that a surprisingly small number of adaptive continuous measurements suffices to accurately recover vectors in high-dimensional spaces, with implications for infinite-dimensional approximation.

Contribution

It introduces a novel measurement strategy that requires only logarithmically many measurements for vector recovery, advancing understanding of measurement efficiency.

Findings

Recovery with $oxed{ ext{logarithmic in } m}$ measurements is possible.

Adaptive measurements outperform non-adaptive methods.

Applications extend to infinite-dimensional approximation problems.

Abstract

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m+1)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.