Dependent latents with infinite correlation exponent

Determine whether approximate message passing can weakly recover the hidden spike v* in the spiked cumulant model when the latent variables (λ, ν) are statistically dependent but have correlation exponent k* = ∞, meaning E[λ^k ν] = 0 for all finite k, in the high-dimensional limit with n, d → ∞ and n = Θ(d).

Background

The paper studies a spiked cumulant model with two latent variables λ and ν. For finite correlation exponent k*, approximate message passing (AMP) and nonlinear autoencoders can weakly recover both spikes. For independent latents with k* = ∞, AMP does not recover v* with n = O(d), aligning with prior hardness results.

However, when λ and ν are dependent yet satisfy E[λk ν] = 0 for all k, the behavior of AMP is not characterized. Establishing if weak recovery is possible in this dependent k* = ∞ regime (and under what sample complexity) would close a gap in the information-theoretic and algorithmic understanding of this model.

References

We finally remark that \cref{res:bo-weak} does not explicitly provide negative results in the case of $k\star = +\infty$. We argue in \cref{app:BO} that if the latent variables are independent, AMP does not weakly recover $\mathbf{v}\star$ for any $n = \mathcal{O}(d)$, as can be expected based on results of . The more subtle case of dependent latents with $k\star = +\infty$ is more elusive, and its analysis is left for future work.

— A solvable high-dimensional model where nonlinear autoencoders learn structure invisible to PCA while test loss misaligns with generalization  (2602.10680 - Mendes et al., 11 Feb 2026) in Section 3 (Information theoretic baseline), end