Norming Sets for Tensor and Polynomial Sketching

2025-06-05

AI Summary

This work addresses the problem of randomized dimensionality reduction for real algebraic varieties and images of polynomial maps—such as low-rank tensors and tensor networks. We develop a unified sketching theory framework for such structured algebraic sets. Our method introduces the *median sketch*, the first approach achieving norm-preserving embedding with only $ ilde{O}(dim V)$ measurements—sharply improving upon the classical $ ilde{O}((dim V)^2)$ bound. The median sketch is compatible with diverse sketching operators, including sub-Gaussian, fast Johnson–Lindenstrauss (JL), and tensor-structured sketches. Leveraging generalized set theory, we provide a unified characterization of their embedding guarantees. Theoretical results ensure efficient compression and accelerated computation for high-dimensional tensors and polynomial-structured data. This yields a tight, scalable foundation for large-scale tensor learning, enabling provably accurate low-dimensional representations while preserving essential algebraic geometry properties.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 extit{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.

Problem

• Develop sketching theory for real algebraic varieties
• Control sketching dimension for various sketch operators
• Propose median sketch method with efficient measurements

Innovation

• Randomized dimension reduction for algebraic varieties
• Norming sets framework for sketching dimension control
• Median sketch method with sparse linear measurements