We study permutation-invariant embeddings of $d$-dimensional point sets, which are defined by sorting $D$ independent one-dimensional projections of the input. Such embeddings arise in graph deep learning where outputs should be invariant to permutations of graph nodes. Previous work showed that for large enough $D$ and projections in general position, this mapping is injective, and moreover satisfies a bi-Lipschitz condition. However, two gaps remain: firstly, the optimal size $D$ required for injectivity is not yet known, and secondly, no estimates of the bi-Lipschitz constants of the mapping are known. In this paper, we make substantial progress in addressing both of these gaps. Regarding the first gap, we improve upon the best known upper bounds for the embedding dimension $D$ necessary for injectivity, and also provide a lower bound on the minimal injectivity dimension. Regarding the second gap, we construct matrices of projection vectors, so that the bi-Lipschitz distortion of the mapping depends quadratically on the number of points $n$, and is completely independent of the dimension $d$. We also show that for any choice of projection vectors, the distortion of the mapping will never be better than a bound proportional to the square root of $n$. Finally, we show that similar guarantees can be provided even when linear projections are applied to the mapping to reduce its dimension.

翻译：我们研究了$d$维点集的置换不变嵌入，该类嵌入通过排序输入数据的$D$个独立一维投影来定义。此类嵌入出现在图深度学习中，其输出应对图节点的置换具有不变性。先前研究表明，当$D$足够大且投影处于一般位置时，该映射是单射的，并且满足双Lipschitz条件。然而，仍存在两个空白：首先，实现单射性所需的最优维度$D$尚不明确；其次，该映射的双Lipschitz常数尚无估计。本文在解决这两个空白方面取得了实质性进展。针对第一个空白，我们改进了实现单射性所需的嵌入维度$D$的已知上界，并给出了最小单射维度的下界。针对第二个空白，我们构造了投影向量矩阵，使得映射的双Lipschitz畸变关于点数$n$呈二次依赖关系，且完全独立于维度$d$。我们还证明，对于任意投影向量选择，映射的畸变不会优于与$\sqrt{n}$成比例的界。最后，我们表明，即使对映射应用线性投影以降低其维度，仍可提供类似的保证。