Optimal Low degree hardness for Broadcasting on Trees

TL;DR

This paper proves that for broadcasting on trees below the Kesten-Stigum bound, any inference method requires polynomial degree complexity, significantly strengthening previous lower bounds and highlighting inherent computational hardness.

Contribution

It establishes an exponential lower bound on the degree of functions needed for root inference, improving prior logarithmic bounds in the entire regime below the Kesten-Stigum threshold.

Findings

Any inference function below the Kesten-Stigum bound requires polynomial degree complexity.

Previous logarithmic lower bounds are exponentially improved.

Root inference complexity is inherently high in the entire sub-Kesten-Stigum regime.

Abstract

Broadcasting on trees is a fundamental model from statistical physics that plays an important role in information theory, noisy computation and phylogenetic reconstruction within computational biology and linguistics. While this model permits efficient linear-time algorithms for the inference of the root from the leaves, recent work suggests that non-trivial computational complexity may be required for inference. The inference of the root state can be performed using the celebrated Belief Propagation (BP) algorithm, which achieves Bayes-optimal performance. Although BP runs in linear time using real arithmetic operations, recent research indicates that it requires non-trivial computational complexity using more refined complexity measures. Moitra, Mossel, and Sandon demonstrated such complexity by constructing a Markov chain for which estimating the root better than random guessing (for typical inputs) is $NC^1$-complete. Kohler and Mossel constructed chains where, for trees with $N$ leaves, achieving better-than-random root recovery requires polynomials of degree $N^{\Omega(1)}$. The papers above raised the question of whether such complexity bounds hold generally below the celebrated Kesten-Stigum bound. In a recent work, Huang and Mossel established a general degree lower bound of $\Omega(\log N)$ below the Kesten-Stigum bound. Specifically, they proved that any function expressed as a linear combination of functions of at most $O(log N)$ leaves has vanishing correlation with the root. In this work, we get an exponential improvement of this lower bound by establishing an $N^{\Omega(1)}$ degree lower bound, for any broadcast process in the whole regime below the Kesten-Stigum bound.