We study representations of data from an arbitrary metric space $\mathcal{X}$ in the space of univariate Gaussian mixtures with a transport metric (Delon and Desolneux 2020). We derive embedding guarantees for feature maps implemented by small neural networks called \emph{probabilistic transformers}. Our guarantees are of memorization type: we prove that a probabilistic transformer of depth about $n\log(n)$ and width about $n^2$ can bi-H\"{o}lder embed any $n$-point dataset from $\mathcal{X}$ with low metric distortion, thus avoiding the curse of dimensionality. We further derive probabilistic bi-Lipschitz guarantees, which trade off the amount of distortion and the probability that a randomly chosen pair of points embeds with that distortion. If $\mathcal{X}$'s geometry is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all points in the dataset. As applications, we derive neural embedding guarantees for datasets from Riemannian manifolds, metric trees, and certain types of combinatorial graphs. When instead embedding into multivariate Gaussian mixtures, we show that probabilistic transformers can compute bi-H\"{o}lder embeddings with arbitrarily small distortion.

翻译：我们从一个任意的计量空间 $\ mathcal{X} 来研究数据的表达方式。 我们从一个任意的测量空间 $\ mathcal{X} 空间里用运输量( Delon 和 Desolneux 2020) 来研究数据。 我们从一个叫做 emph{ 概率变异器的小型神经网络所执行的地貌图的嵌入保证。 我们的保证是记忆型的: 我们证明一个深度约 $n\log(n) 和宽度约 $n%2$ 的概率变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变性变异性变异性变异性变异性变异性变性变性变异性变性变型的模型。