发表机构
Computer Science and Mathematics Division, Oak Ridge National Laboratory（计算机科学与数学分部，橡树岭国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究揭示高斯模型中Johnson-Lindenstrauss（JL）界无法反映保留的几何信息，推导了随机投影恢复特征的奇异值极限，得出其在距离恢复、近邻排名及协方差形状上的精确结论。
AI 中文摘要
Johnson-Lindenstrauss（JL）引理保证，将n个点随机投影到m=O(ε⁻²log n)维空间时，可高概率将两两平方距离保持在相对误差ε内，且该维度阶数是渐近最优的。然而在高维场景中，距离会集中于基线，而关键几何信息则包含在小得多的波动中。我们证明JL界可能无法反映保留的几何：独立高斯替换映射可满足JL界，即便替换点云与原始数据无关。随后我们研究任意解码器从线性草图中恢复平方距离D的特征f(D)的性能，在平方误差损失下，最优解码器为条件期望，故恢复过程定义了一个线性算子，其奇异值可量化特征恢复效果。对于各向同性高斯数据（Σ=σ²I_d），我们对该算子进行了闭式对角化；当固定k且m、d−m→∞时，其第k个奇异值满足ℓ_k≈(m/d)^(k/2)。这产生三个明确结论：秩为m的草图最多保留任意单个平方距离特征方差的m/d比例；若m→∞且m/d→0，期望Kendall相关系数为(2/π)√(m/d)(1+o(1))；对于固定q，近邻一致性趋于1/q。此外，当log n≪m≪d时，单个投影可满足JL界，同时平均Kendall相关系数消失；去除尺度后，Haar平均保留的协方差形状信息为(m/d)²。因此，JL距离保持无法量化用于比较或推断的几何信息。
英文摘要
The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($Σ=σ^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}π\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.
Comments41 pages, 6 figures, 4 tables. Reproducibility code: https://github.com/piyush314/random-projection-geometry (pinned as a submodule). Companion paper on nearest-neighbor graphs to follow
