How many continuous measurements are needed to learn a vector?

TL;DR

The paper shows that recovering a vector $x\in\mathbb{R}^m$ from adaptive continuous measurements can be done with far fewer than $m$ measurements, specifically $n(m)=\lceil\log_2(m+1)\rceil+1$, by constructing a partition of $\mathbb{R}^m$ into $m+1$ color regions and using distance-to-set measurements plus a Lipschitz selector. The main contribution is a concrete algorithm $R_m^\varepsilon$ that achieves $\|x-\hat{x}\|\le\varepsilon$ with $n(m)$ adaptive measurements, together with a careful partition and selection mechanism that enables binary search over colors and disambiguation within a color class. Beyond the finite-dimensional result, the authors connect adaptive continuous information to manifold widths, showing that adaptive measurements yield exponential speed-ups over non-adaptive ones and deriving general bounds that extend to infinite-dimensional settings. They also discuss implications for Lipschitz measurements, provide insights into the Hilbert-space case where widths align and adaptivity is particularly powerful, and outline open questions about optimality and broader classes of continuous information.

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.

Figures (1)

• Figure 1: Two colorings of the plane with three colors. The second is generalized above to higher dimensions.

Theorems & Definitions (12)

• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 6
• proof
• Lemma 7
• proof
• Theorem 8