Compressive learning is an approach to efficient large scale learning based on sketching an entire dataset to a single mean embedding (the sketch), i.e. a vector of generalized moments. The learning task is then approximately solved as an inverse problem using an adapted parametric model. Previous works in this context have focused on sketches obtained by averaging random features, that while universal can be poorly adapted to the problem at hand. In this paper, we propose and study the idea of performing sketching based on data-dependent Nystr\"om approximation. From a theoretical perspective we prove that the excess risk can be controlled under a geometric assumption relating the parametric model used to learn from the sketch and the covariance operator associated to the task at hand. Empirically, we show for k-means clustering and Gaussian modeling that for a fixed sketch size, Nystr\"om sketches indeed outperform those built with random features.

翻译：压缩学习是高效大规模学习的一种方法,其基础是将整个数据集素描成单一平均嵌入(素描),即通用瞬间矢量。学习任务随后大约作为一个反问题用一个经调整的参数模型解决。以前在这方面的工作侧重于通过平均随机特征获得的素描,虽然通用性不适应手头的问题。在本文中,我们提议并研究根据依赖数据的Nystr\'om近似进行素描的想法。从理论角度看,我们证明在与手头任务相关联的参数模型和同源操作器有关的参数模型的几何假设下,可以控制过重的风险。有规律的,我们用K- means集群和高斯模型显示,对于固定的素描尺寸,Nystr\'om的素描图确实比随机特征所建的模型要强。