Latent tree graphical models are widely used in computational biology, signal and image processing, and network tomography. Here we design a new efficient, estimation procedure for latent tree models, including Gaussian and discrete, reversible models, that significantly improves on previous sample requirement bounds. Our techniques are based on a new hidden state estimator which is robust to inaccuracies in estimated parameters. More precisely, we prove that latent tree models can be estimated with high probability in the so-called Kesten-Stigum regime with $O(log^2 n)$ samples where $n$ is the number of nodes.

The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixtureof phylogenies. Such mixed models play an increasingly important role in practice. Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.

Mutation rate variation across loci is well known to cause difficulties, notably identifiability issues, in the reconstruction of evolutionary trees from molecular sequences. Here we introduce a new approach for estimating general rates-across-sites models. Our results imply, in particular, that large phylogenies are typically identifiable under rate variation. We also derive sequence-length requirements for high-probability reconstruction. Our main contribution is a novel algorithm that clusters sites according to their mutation rate. Following this site clustering step, standard reconstruction techniques can be used to recover the phylogeny. Our results rely on a basic insight: that, for large trees, certain site statistics experience concentration-of-measure phenomena.

We present an efficient phylogenetic reconstruction algorithm, allowing insertions and deletions, which provably achieves a sequence-length requirement (or sample complexity) growing polynomially in the number of taxa. Our algorithm is distance-based, that is, it relies on pairwise sequence comparisons. More importantly, our approach largely bypasses the difficult problem of multiple sequence alignment.

We consider the trace reconstruction problem on a tree (TRPT): a binary sequence is broadcast through a tree channel where we allow substitutions, deletions, and insertions; we seek to reconstruct the original sequence from the sequences received at the leaves. The TRPT is motivated by the multiple sequence alignment problem in computational biology. We give a simple recursive procedure giving strong reconstruction guarantees at low mutation rates.

We begin with a random sequence $x_1, \ldots, x_k$ which is located at the root. If vertex $v$ has the sequence $y_1, \ldots y_{k_{v}}$, then each one of its $d$ children will have a sequence which is generated from $y_1, \ldots y_{k_{v}}$ by flipping three biased coins for each bit. The first coin has probability $p_s$ for Heads, and determines whether this bit will be substituted or not. The second coin has probability $p_d$, and determines whether this bit will be deleted, and the third coin has probability $p_i$ and determines whether a new random bit will be inserted. The input to the procedure is the sequences of the $n$ leaves of the tree, as well as the tree structure (but not the sequences of the inner vertices) and the goal is to reconstruct an approximation to the sequence of the root (the DNA of the ancestral father). For every $\epsilon > 0$, we present a deterministic algorithm which outputs an approximation of $x_1, \ldots, x_k$ with probability $1 - \epsilon$, if $p_i + p_d < O(1/k^{2/3} \log n)$ and $(1 - 2p_s)^2 > O(d^{-1} \log d)$, where the probability is on the coin flips which determine the mutation process.

To our knowledge, this is the first rigorous trace reconstruction result on a tree in the presence of indels.