Random restrictions of high-rank tensors and polynomial maps

TL;DR

This paper investigates how random coordinate restrictions affect the rank of tensors and polynomial maps, showing that for many rank functions, the rank decreases only modestly under such restrictions.

Contribution

It introduces the concept of natural rank functions and demonstrates that random restrictions typically preserve most of the rank, advancing understanding in computational complexity.

Findings

Rank decreases by at most a constant factor under dense random restrictions

Natural rank functions are stable under random coordinate restrictions

Provides bounds on rank reduction for broad classes of rank functions

Abstract

Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank functions, random coordinate restriction to a dense set will typically reduce the rank by at most a constant factor.