Let X _v for v ∈V be a family of n iid uniform points in the square . Suppose first that we are given the random geometric graph , where vertices u and v are adjacent when the Euclidean distance d _E (X _u ,X _v ) is at most r . Let n ^3/14≪r ≪n ^1/2. Given G (without geometric information), in polynomial time we can with high probability approximately reconstruct the hidden embedding, in the sense that “up to symmetries,” for each vertex v we find a point within distance about r of X _v ; that is, we find an embedding with “displacement” at most about r . Now suppose that, instead of G we are given, for each vertex v , the ordering of the other vertices by increasing Euclidean distance from v . Then, with high probability, in polynomial time we can find an embedding with displacement .

中文翻译：

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从几何图或顺序中学习随机点

让X _v为v ∈ V是一个家庭ñ广场IID均匀点。首先假设给定随机几何图，当欧几里得距离d _E（X _u，X _v）最多为r时，顶点u和v相邻。让ñ ^3/14 « [R « ñ ^1/2。给定G（没有几何信息），在多项式时间内，我们可以高概率近似地重构隐藏的嵌入，即“达到对称性”，对于每个顶点v，我们在x _v的r的距离内找到一个点；也就是说，我们发现关于r最多具有“位移”的嵌入。现在假设，而不是摹我们得到，对于每一个顶点v，其他顶点通过增加欧几里得距离排序v。然后，在多项式时间内，我们很有可能找到一个带位移的嵌入。