The minimax rate for estimating d-th order moment tensors is sqrt(p/n) wedge 1, while low-degree evidence shows detection of vanishing cumulants is hard for n much less than p to the d/2, creating a reverse detection-estimation gap.
Tensor-on-tensor regression: Riemannian optimization, over-parameterization, statistical-computational gap and their interplay.The Annals of Statistics, 52(6):2583–2612, 2024
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Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors
The minimax rate for estimating d-th order moment tensors is sqrt(p/n) wedge 1, while low-degree evidence shows detection of vanishing cumulants is hard for n much less than p to the d/2, creating a reverse detection-estimation gap.