Optimal Low degree hardness for Broadcasting on Trees

TL;DR

This work analyzes root-reconstruction in the Broadcast on Trees (BOT) model through a low-degree framework, showing that below the KS threshold ($d\lambda^2<1$) any polynomial of the leaves with Efron–Stein degree up to $\exp\left( c\ell/(\log R+1) \right)$ has vanishing correlation with the root, despite Belief Propagation achieving linear-time recovery. The authors develop an extensive operator- and tensor-norm toolkit, introducing R-spaces and a two-part inductive decomposition that separates high-degree polynomials into tractable, degree-1 components and higher-degree remnants. The main technical contribution is an exponential improvement over prior bounds, demonstrating that the KS boundary is a sharp transition between feasible low-degree estimators and exponential-degree hardness, effectively tying computational hardness to network depth analogies in neural networks. The results hold for general ergodic Markov chains below KS, providing a rigorous hardness certificate for a broad class of BOT channels and offering a depth-based perspective on why BP remains efficient while shallow, bounded-degree architectures fail to replicate its performance. These insights deepen the connection between statistical physics, information theory, and computational complexity in structured random models and have implications for understanding depth versus width in algorithmic inference on trees.

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 $N^{Ω(1)}$. 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 $Ω(\log N)$ 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 $N^{Ω(1)}$ degree lower bound, for any broadcast process in the whole regime below the Kesten-Stigum bound.

Figures (11)

• Figure 1: An example of a binary rooted tree of depth 5 is shown. The vertex $u$ is at the 3rd layer and ${\rm h}(u)=2$. Further, the following relationships hold: $v < u$ and $v$ is a child of $u$, $s \in L$ is a 2nd descendant of $u$, $w$ is the parent of $u$, and $t$ is the 2nd ancestor of $u$.
• Figure 2: Both $f_u$ and $f_v$ are in variables $x_{L_u}$ and $x_{L_v}$, respectively. On the other hand, $f_u$ has higher degrees in $x_{L_u}$ and $f_v$ has higher degrees in $x_{L_v}$.
• Figure 3: A function of degree $K+1$ with variable $u,u_1,\dots, u_K$ which cannot be handled by prior work HanMossel:23 when $K$ is beyond $O(\log(N))$.
• Figure 4: An illustration of the sets $A(u;k)$.
• Figure 5: An illustration of the definition of ${\mathcal{R}}({{\mathcal{W}}_u},k)$ for $k=0,1,2$.

Theorems & Definitions (124)

• Theorem : Informal
• Definition 1.1
• Definition 1.2
• Remark 1.3: Markov Property
• Definition 1.4: Efron-Stein Degree
• Theorem 1.5
• Remark 1.6
• Remark 1.7: Variance Decay implies Vanishing Correlation with Root
• Definition 1.8
• Definition 2.1: Antichains