A Framework for Computational Lower Bounds in Nontrivial Norm Approximation

TL;DR

This paper introduces a framework that converts detection-estimation gaps into lower bounds for approximating nontrivial norms, demonstrated on tensor spectral norm with implications for computational hardness.

Contribution

It develops a general method to certify the hardness of norm approximation by leveraging reverse detection-estimation gaps and applies it to tensor spectral norm.

Findings

Any low-degree algorithm with D ≤ c_d(log p)^2 must incur high distortion.

Under the low-degree conjecture, the same hardness extends to all polynomial-time algorithms.

The lower bound matches known upper bounds up to polylogarithmic factors, indicating a genuine computational barrier.

Abstract

In this note, we propose a 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.