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.

翻译：本文提出了一个框架，通过利用反向检测-估计间隙来证明范数近似中的计算下界。起点是一个检验问题，同时存在一个估计器，其误差显著小于相应的计算检测阈值。我们证明，这种间隙为该计算类中任何算法可达到的近似失真度提供了一个下界。这样一来，反向检测-估计间隙可转化为一种通用机制，用于确认非平凡范数近似的难度。我们将此框架应用于$\mathbb{R}^{p^d}$中$d$阶对称张量的谱范数。利用最近建立的用于检测非零高阶累积量张量的低度困难结果，结合一个误差低于低度检测阈值的可高效计算估计器，我们证明任何度数$D \le c_d(\log p)^2$的低度算法在张量谱范数下必须产生至少$p^{d/4-1/2}/\operatorname{polylog}(p)$的失真度。在低度猜想下，该结论可推广至所有多项式时间算法。在若干重要场景中，此下界与已知最佳上界相匹配（仅差多对数因子），表明指数$d/4-1/2$刻画了真正的计算障碍。我们的结果证明，张量谱范数近似的困难性并非现有技术的产物，而是反映了更广泛的计算壁垒。