A General Framework for Computational Lower Bounds in Nontrivial Norm Approximation

2026-04-01

AI Summary

This work investigates computational lower bounds for approximating nontrivial norms of higher-order tensors, such as the spectral norm. We introduce a general framework that systematically formalizes the detection–estimation gap as a mechanism for establishing computational hardness, realized through the low-degree polynomial method under the low-degree conjecture. Applying this framework to the symmetric tensor spectral norm, we prove that any degree-$D$ algorithm with $D \leq c_d(\log p)^2$ incurs an approximation distortion of at least $p^{d/4 - 1/2}/\mathrm{polylog}(p)$. This lower bound matches existing upper bounds up to polylogarithmic factors in several important regimes, thereby revealing an inherent computational barrier characterized by the exponent $d/4 - 1/2$.

Abstract

In this note, we propose a general framework for proving computational lower bounds in norm approximation by leveraging a reverse detection--estimation gap. The starting point is a testing problem together with an estimator whose error is significantly smaller than the corresponding computational detection threshold. We show that such a gap yields a lower bound on the approximation distortion achievable by any algorithm in the underlying computational class. In this way, reverse detection--estimation gaps can be turned into a general mechanism for certifying the hardness of approximating nontrivial norms. We apply this framework to the spectral norm of order-$d$ symmetric tensors in $\mathbb{R}^{p^d}$. Using a recently established low-degree hardness result for detecting nonzero high-order cumulant tensors, together with an efficiently computable estimator whose error is below the low-degree detection threshold, we prove that any degree-$D$ low-degree algorithm with $D \le c_d(\log p)^2$ must incur distortion at least $p^{d/4-1/2}/\operatorname{polylog}(p)$ for the tensor spectral norm. Under the low-degree conjecture, the same conclusion extends to all polynomial-time algorithms. In several important settings, this lower bound matches the best known upper bounds up to polylogarithmic factors, suggesting that the exponent $d/4-1/2$ captures a genuine computational barrier. Our results provide evidence that the difficulty of approximating tensor spectral norm is not merely an artifact of existing techniques, but reflects a broader computational barrier.

Problem

• norm approximation
• computational lower bounds
• tensor spectral norm
• computational hardness
• detection–estimation gap

Innovation

• computational lower bounds
• reverse detection–estimation gap
• tensor spectral norm
• low-degree polynomial hardness
• norm approximation