Reconstruction on Trees and Low-Degree Polynomials

The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, MCMC algorithms, and community detection in random graphs. Notably, the celebrated Belief Propagation (BP) algorithm achieves optimal performance for the reconstruction problem of predicting the value of the Markov process at the root of the tree from its values at the leaves. Recently, the analysis of low-degree polynomials has emerged as a valuable tool for predicting computational-to-statistical gaps. In this work, we investigate the performance of low-degree polynomials for the reconstruction problem. Perhaps surprisingly, we show that there are simple tree models of fixed arity $d$ and growing depth $\ell$ (so $N = 2^{\ell \log_2(d)}$ leaves) where (1) nontrivial reconstruction of the root value is possible with a simple polynomial time algorithm and with robustness to noise, but not with any polynomial of degree $2^{c \ell} = N^{c/\log_2(d)}$ for $c > 0$ a constant, and (2) when the tree is unknown and given multiple samples with correlated root assignments, nontrivial reconstruction of the root value is possible with a simple, noise-robust, and computationally efficient SQ algorithm but not with any polynomial of degree $2^{c \ell}$. These results clarify limitations of low-degree polynomials vs. polynomial time algorithms for Bayesian estimation problems. They also complement recent work of Moitra, Mossel, and Sandon who studied the circuit complexity of Belief Propagation. As a consequence of our main result, we show that for some $c' > 0$, $\exp(2^{c'\ell}) = \exp(N^{c'/\log_2(d)})$ many samples are needed for RBF kernel regression to obtain nontrivial correlation with the true regression function (BP). We pose related open questions about low-degree polynomials and the Kesten-Stigum threshold.