Low-Rank Tensor Recovery via Theta Nuclear p-Norm

TL;DR

This work develops a theta-body relaxation framework for low-rank tensor recovery via nuclear $p$-norms. It provides algebraic descriptions through ideals $I_p$ and their reduced Gröbner bases, enabling semidefinite programming relaxations that approximate the nuclear $p$-norm unit ball. The geometry of theta bodies is analyzed, establishing real radicality for key cases and theta-exactness results, and revealing symmetry and duality structures that aid certificate construction. The paper also derives a Gaussian-width-based lower bound suggesting $m_0=O(n)$ measurements suffice for exact recovery in certain tensor regimes, supported by numerical experiments comparing theta-1 relaxations for different $p$-norms. Collectively, these results offer a principled SDP approach to tensor recovery with scalable measurement requirements and concrete algebraic underpinnings.

Abstract

We investigate the low-rank tensor recovery problem using a relaxation of the nuclear p-norm by theta bodies. We provide algebraic descriptions of the norms and compute their Gröbner bases. Moreover, we develop geometric properties of these bodies. Finally, our numerical results suggest that for $n\times\cdots\times n$ tensors, $m\geq O(n)$ measurements should be sufficient to recover low-rank tensors via theta body relaxation.

Figures (5)

• Figure 1: Comparison of number of measurements required
• Figure 2: Computing time comparison
• Figure 4: Number of measurements required by $\mathop{\mathrm{TH}}\nolimits_{1}(I_\infty)$ and $\mathop{\mathrm{TH}}\nolimits_{1}(I_2)$ on general tensors
• Figure 5: $n\times n\times n$ tensors
• Figure 6: $n\times n\times n\times n$ tensors

Theorems & Definitions (84)

• Theorem 1
• Theorem 2
• Conjecture 3
• Definition 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Definition 1.5: Spectrahedron and semidefinite program
• Theorem 1.6
• Theorem 1.7