Extension of the generic-chaining robust tensor estimator to optimal corruption rates beyond fourth moments

Determine whether the generic-chaining framework based on admissible sequences of nets and robust one-dimensional mean estimation extends to achieve optimal corruption rates for robust simple-tensor estimation under moment assumptions with p>4.

Background

The paper compares its estimator with the robust simple-tensor estimator of Bartl and Mendelson. Under adversarial contamination, that prior construction establishes the optimal corruption rate only under fourth-moment assumptions. Its methodology combines generic chaining over an admissible sequence of nets with robust one-dimensional mean estimation.

The unresolved issue is whether this framework can be extended to moment parameters p>4 while retaining optimal dependence on the contamination rate. The present paper claims to resolve the broader statistical objective through a different estimator based on directional trimmed means and minimax aggregation, but explicitly identifies the extension of the earlier framework itself as open.

References

Under adversarial contamination, however, their optimal corruption rate is established only under fourth-moment assumptions. Their construction combines generic chaining through an admissible sequence of nets with robust one-dimensional mean estimation, and whether this framework extends to the optimal corruption rates for $p>4$ remains open.

— Robust dimension-free estimation of simple random tensors: optimal guarantees under heavy tails and adversarial contamination  (2609.00675 - Oliveira et al., 1 Sep 2026) in Section 1, subsection “Comparison with prior robust tensor estimators” (Section 1.2)