Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings
基于排序的置换不变嵌入的定量界
Nadav Dym, Matthias Wellershoff, Efstratios Tsoukanis, Daniel Levy, Radu Balan
AI总结 研究通过排序独立一维投影得到的置换不变嵌入,改进了注入性所需嵌入维度的上下界,并给出了双Lipschitz常数的估计,其失真度与点数n的平方成正比且与维度d无关。
Comments Minor revision; 37 pages, 1 figure, 2 tables
详情
- Journal ref
- IEEE Trans. Inf. Theory, vol. 72, no. 6, pp. 4297-4311, Jun. 2026
我们研究$d$维点集的置换不变嵌入,这些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对图节点输出应具有置换不变性的场景。先前的工作表明,对于足够大的$D$和处于一般位置的投影,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,注入性所需的最优大小$D$尚不清楚;其次,映射的双Lipschitz常数估计未知。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,我们改进了注入性所需嵌入维度$D$的最佳已知上界,并给出了最小注入性维度的下界。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失真度与点数$n$的平方成正比,且完全独立于维度$d$。我们还证明,对于任何投影向量的选择,映射的失真度不会优于与$n$的平方根成比例的界。最后,我们展示了即使对映射应用线性投影以降低其维度,也能提供类似的保证。
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.