The Real Vanishing Ideals of Nuclear p-Norm Balls

TL;DR

This work links convex geometry and real algebraic geometry in the study of tensor nuclear norms by showing the unit ball of the nuclear norm is the convex hull of a real irreducible variety defined by a real vanishing ideal. It provides a practical prime-ideal criterion that reduces primeness verification to elimination polynomials and strong square-freeness, and applies it to the nuclear p-norm ideals I_p, proving real radicalness for p ∈ {0,1,2,∞} and, in particular, that I_2 is real reduced and prime. The authors also characterize the I_p as G-invariant, with the real variety essentially the orbit of a rank-one tensor under an orthogonal group action, and they establish primeness and real-reducedness via a combination of complexification, symmetry arguments, and a single non-singular real point argument. This provides a solid algebraic-geometry foundation for analyzing nuclear-norm relaxations and related theta-body constructions in tensor recovery and low-rank optimization.

Abstract

We study the algebraic and geometric structure related to tensor nuclear norms. We show that the unit ball of the nuclear norm is the convex hull of an irreducible real variety and give an explicit description of its real vanishing ideal. As a consequence, we obtain a simple criterion to decide when a primary ideal is prime, and we use it to prove that the ideal of the nuclear 2-norm is real reduced and prime.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2: Nullstellensatz
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof