Johnson–Lindenstrauss引理中的精确维数边界
The Sharp Dimension Bound in the Johnson--Lindenstrauss Lemma
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中文总结 AI 辅助
本文证实了Larsen和Nelson关于Johnson–Lindenstrauss引理最优目标维数的猜想，证明该上界可通过线性映射达到，且匹配下界对非线性嵌入也成立。
中文摘要 AI 辅助
Johnson–Lindenstrauss引理断言，d维欧氏空间中任意n个点的集合都可嵌入到O(ε⁻²log n)维欧氏空间，且失真不超过1+ε。Larsen和Nelson猜想，在参数n、d、ε的整个取值范围内，最优目标维数为Θ(min{d, n−1, log(2+ε²n)/ε²})。我们证实了该猜想成立，且进一步证明该上界可通过线性映射达到；而匹配的下界（由Larsen–Nelson和Alon–Klartag给出）甚至对非线性嵌入也成立。
英文摘要
The Johnson--Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d, \varepsilon$ is \[ Θ\left(\min\left\{d,n-1,\frac{\log(2+\varepsilon^2n)}{\varepsilon^2}\right\}\right). \] We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen--Nelson and Alon--Klartag, holds even for nonlinear embeddings.
