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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.26029 |
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| _version_ | 1866910102372483072 |
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| author | Tang, Runshi Han, Yuefeng Zhang, Anru R. |
| author_facet | Tang, Runshi Han, Yuefeng Zhang, Anru R. |
| contents | We study estimation and detection of high-order moment and cumulant tensors from $n$ i.i.d.\ observations of a $p$-dimensional random vector, with performance measured in tensor spectral norm. Under sub-Gaussianity, we show that the minimax rate for estimating the order-$d$ moment and cumulant tensors is $\sqrt{p/n}\wedge 1$. In contrast to covariance estimation, the sample moment tensor is generally not rate-optimal for $d\ge 3$, and we construct an estimator that attains the minimax rate up to logarithmic factors. On the computational side, we study testing whether the $d$-th order cumulant tensor vanishes after whitening. Using the low-degree polynomial framework, we provide evidence that detection is computationally hard when $n\ll p^{d/2}$. At the same time, we identify a regime in which an efficiently computable estimator achieves error smaller than the separation at which low-degree tests can reliably distinguish the null from the alternative. This reveals an unusual reverse detection--estimation gap: computationally efficient detection can be harder than computationally efficient estimation. The underlying reason is that the relevant loss, tensor spectral norm, is itself NP-hard to compute, creating a new form of computational--statistical gap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors Tang, Runshi Han, Yuefeng Zhang, Anru R. Statistics Theory Computational Complexity We study estimation and detection of high-order moment and cumulant tensors from $n$ i.i.d.\ observations of a $p$-dimensional random vector, with performance measured in tensor spectral norm. Under sub-Gaussianity, we show that the minimax rate for estimating the order-$d$ moment and cumulant tensors is $\sqrt{p/n}\wedge 1$. In contrast to covariance estimation, the sample moment tensor is generally not rate-optimal for $d\ge 3$, and we construct an estimator that attains the minimax rate up to logarithmic factors. On the computational side, we study testing whether the $d$-th order cumulant tensor vanishes after whitening. Using the low-degree polynomial framework, we provide evidence that detection is computationally hard when $n\ll p^{d/2}$. At the same time, we identify a regime in which an efficiently computable estimator achieves error smaller than the separation at which low-degree tests can reliably distinguish the null from the alternative. This reveals an unusual reverse detection--estimation gap: computationally efficient detection can be harder than computationally efficient estimation. The underlying reason is that the relevant loss, tensor spectral norm, is itself NP-hard to compute, creating a new form of computational--statistical gap. |
| title | Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors |
| topic | Statistics Theory Computational Complexity |
| url | https://arxiv.org/abs/2603.26029 |