Zhixin Zhou and Yizhe Zhu, in an arXiv preprint, prove sharp spectral-norm bounds for sparse random tensors of fixed order k. For an n x ... x n tensor with independent Bernoulli(p) entries, the centered spectral norm is O(sqrt(np)) with polynomially high probability whenever np >= c log n. This removes the logarithmic loss from the authors' 2021 result.
At every sparsity level 0 < p <= 1/2, the expected centered norm is of order min{n^k p, sqrt(d_eff)}, where d_eff is an effective degree that accounts for unusually large fiber counts. The upper bounds also extend to tensors with deterministic entrywise weights and unequal sampling probabilities, and analogous two-sided bounds hold for the Friedman–Wigderson model at every degree.
For independent Bernoulli tensors with unknown, possibly unequal means in [0,p], the paper introduces an all-sparsity regularization procedure. The light-fitting method estimates the original mean with minimax expected spectral-norm error of order min{sqrt(np), p n^{k/2}} for every known p in [0,1]. Instead of deleting high-degree fibers, it fits light multilinear residuals under an entrywise constraint on the candidate mean.