高维几何植入匹配：多视图的力量
Geometric planted matchings in high dimensions: The power of multiple views
浏览论文内容
中文总结 AI 辅助
研究\(\mathbb{R}^d\)中\(n\)个点与噪声置换版本的对应关系恢复问题，高维下\(b = 2\)是阈值。证明\(b < 2\)时无估计器能有效恢复。多视图推广中，\(b > K/(K - 1)\)时简单过程可恢复匹配，打破\(b = 2\)障碍。
中文摘要 AI 辅助
我们研究在\(\mathbb{R}^d\)中\(n\)个点的集合与这些点的噪声、置换版本之间恢复对应关系的问题。在高维情形\(d = \omega(\log n)\)下，在噪声方差\(\sigma^2 = d/(b\log n)\)的高斯模型中，先前工作确定\(b = 2\)为几乎精确恢复的阈值。我们证明此阈值是全有或全无的：对于每个固定的\(b < 2\)，没有估计器能恢复匹配的正分数，甚至在欧几里得距离中估计匹配点云渐近上也不比忽略对应关系更好。另一方面，我们考虑该问题的多视图推广，其中观察到同一潜在点云的\(K\)个噪声、独立置换副本。在此我们表明，当\(b > K/(K - 1)\)时，一个简单的多项式时间过程能恢复所有相对匹配，误差至多为\(o(n)\)。因此多视图可以打破原始匹配问题的\(b = 2\)的不可能障碍：特别是，对于\(3/2 < b < 2\)，双视图模型没有非平凡恢复，但第三个视图使所有潜在对应关系可有效恢复。
英文摘要
We study the problem of recovering the correspondence between a collection of $n$ points in $\mathbb{R}^d$ and a noisy, permuted version of those points. In the high-dimensional regime $d=ω(\log n)$, under a Gaussian model with noise variance $σ^2=d/(b\log n)$, prior work identifies $b=2$ as the threshold for almost exact recovery. We prove that this threshold is all-or-nothing: for every fixed $b<2$, no estimator recovers a positive fraction of the matching, and even estimating the matched point cloud in Euclidean distance is asymptotically no better than ignoring the correspondence. On the other hand, we consider a multi-view generalization of the problem where $K$ noisy, independently permuted copies of the same latent point cloud are observed. Here we show that a simple polynomial-time procedure recovers all relative matchings up to $o(n)$ errors whenever $b>K/(K-1)$. Thus multiple views can break the impossibility barrier $b=2$ for the original matching problem: in particular, for $3/2 < b < 2$, the two-view model has no nontrivial recovery, but a third view makes all latent correspondences efficiently recoverable.
