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In this paper, we study the estimation of a rank-one spiked tensor in thepresence of heavy tailed noise. Our results highlight some of the fundamentalsimilarities and differences in the tradeoff between statistical andcomputational efficiencies under heavy tailed and Gaussian noise. Inparticular, we show that, for $p$ th order tensors, the tradeoff manifests inan identical fashion as the Gaussian case when the noise has finite $4(p-1)$ thmoment. The difference in signal strength requirements, with or withoutcomputational constraints, for us to estimate the singular vectors at theoptimal rate, interestingly, narrows for noise with heavier tails and vanisheswhen the noise only has finite fourth moment. Moreover, if the noise has lessthan fourth moment, tensor SVD, perhaps the most natural approach, issuboptimal even though it is computationally intractable. Our analysis exploitsa close connection between estimating the rank-one spikes and the spectral normof a random tensor with iid entries. In particular, we show that the order ofthe spectral norm of a random tensor can be precisely characterized by themoment of its entries, generalizing classical results for random matrices. Inaddition to the theoretical guarantees, we propose estimation procedures forthe heavy tailed regime, which are easy to implement and efficient to run.Numerical experiments are presented to demonstrate their practical merits.

On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors