Quantitative Bounds for Sorting based Permutation Invariant Embeddings

Radu Balan, Nadav Dym, Daniel Levy, Efstratios Tsoukanis, Matthias Wellershoff

Session: Invariant theory for machine learning (Dustin Mixon, Soledad Villar)

Keywords: sorting, permutation invariants, bi-Lipschitz maps

Abstract:

In this paper we analyze injective Euclidean embeddings of the metric space $\mathbb{R}^{n\times d}/\sim$, where $X\sim Y$ if and only if there is a permutation matrix $P$ so that $Y=PX$. We construct embeddings $\beta_A:\mathbb{R}^{n\times d}\rightarrow \mathbb{R}^{n\times D}$ and $\gamma_{A,B}:\mathbb{R}^{n\times d}\rightarrow \mathbb{R}^m$ of the form $X\mapsto \beta_A(X)=sort(XA)$ and
$X\mapsto \gamma_{A,B}(X)=diag(B^T sort(XA))$ for pairs of matrices $(A,B)$. We show that generic $A$ and $(A,B)$ of size $D=n(d-1)+1$ and $m=2nd-d$ produce injective embeddings. For $D=O(n^2 d)$ we obtain that, with high probability,
$\beta_A$ is a bi-Lipschitz map with distortion bounded by $O(n^2\sqrt{d})$.

Submission Number: 31