Norming Sets for Tensor and Polynomial Sketching

TL;DR

This paper introduces a theoretical framework for tensor and polynomial sketching using norming sets, enabling efficient dimension reduction for complex algebraic varieties and tensor networks with various sketching operators.

Contribution

It develops a norming set-based theory for sketching algebraic varieties and proposes the median sketch method for efficient embedding with fewer measurements.

Findings

Median sketch uses $ ilde{O}( ext{dim }V)$ measurements.

Framework applies to sub-Gaussian, Johnson-Lindenstrauss, and tensor structured sketches.

Provides bounds on sketching dimensions for low-rank tensors and polynomial images.

Abstract

This paper develops the sketching (i.e., randomized dimension reduction) theory for real algebraic varieties and images of polynomial maps, including, e.g., the set of low rank tensors and tensor networks. Through the lens of norming sets, we provide a framework for controlling the sketching dimension for \textit{any} sketch operator used to embed said sets, including sub-Gaussian, fast Johnson-Lindenstrauss, and tensor structured sketch operators. Leveraging norming set theory, we propose a new sketching method called the median sketch. It embeds such a set $V$ using only $\widetilde{\mathcal{O}}(\dim V)$ tensor structured or sparse linear measurements.