Mean Nystr\"om Embeddings for Adaptive Compressive Learning

TL;DR

This paper introduces a data-dependent Nyström method for sketching in compressive learning, demonstrating improved performance over random feature methods in clustering and modeling tasks through theoretical analysis and empirical validation.

Contribution

It proposes and analyzes a Nyström-based sketching approach for compressive learning, offering theoretical guarantees and empirical evidence of its advantages over random feature sketches.

Findings

Nyström sketches outperform random feature sketches in clustering and modeling.

Theoretical excess risk bounds are established under geometric assumptions.

Empirical results confirm improved accuracy with fixed sketch size.

Abstract

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.