Efficient Projection onto the Perfect Phylogeny Model

TL;DR

This paper introduces a fast, exact algorithm for projecting onto the perfect phylogeny model, enabling efficient exploration of evolutionary trees with up to 10 nodes, which is valuable for biological inference.

Contribution

The paper presents a novel algorithm using Moreau's decomposition and tree reduction for exact, rapid projection onto the perfect phylogeny model.

Findings

Algorithm terminates with an exact solution in finite steps.

Can explore over 2 billion trees with fewer than 11 nodes in about 2 hours.

Significantly faster than existing methods for this problem.

Abstract

Several algorithms build on the perfect phylogeny model to infer evolutionary trees. This problem is particularly hard when evolutionary trees are inferred from the fraction of genomes that have mutations in different positions, across different samples. Existing algorithms might do extensive searches over the space of possible trees. At the center of these algorithms is a projection problem that assigns a fitness cost to phylogenetic trees. In order to perform a wide search over the space of the trees, it is critical to solve this projection problem fast. In this paper, we use Moreau's decomposition for proximal operators, and a tree reduction scheme, to develop a new algorithm to compute this projection. Our algorithm terminates with an exact solution in a finite number of steps, and is extremely fast. In particular, it can search over all evolutionary trees with fewer than 11 nodes, a size relevant for several biological problems (more than 2 billion trees) in about 2 hours.