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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.00966 |
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| _version_ | 1866914444633702400 |
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| author | Tang, Runshi Han, Yuefeng Zhang, Anru R. |
| author_facet | Tang, Runshi Han, Yuefeng Zhang, Anru R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00966 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Framework for Computational Lower Bounds in Nontrivial Norm Approximation Tang, Runshi Han, Yuefeng Zhang, Anru R. Statistics Theory Computational Complexity 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. |
| title | A Framework for Computational Lower Bounds in Nontrivial Norm Approximation |
| topic | Statistics Theory Computational Complexity |
| url | https://arxiv.org/abs/2604.00966 |