Tree construction using singular value decomposition

We present a new, statistically consistent algorithm for phylogenetic tree construction that uses the algebraic theory of statistical models (as developed in [5]). Our basic tool is the Singular Value Decomposition (SVD) from numerical linear algebra (see [3]). Starting with a multiple alignment of n species, we show that the SVD allows us to decide whether a split of the species occurs in their phylogenetic tree. Using this fact, we have developed an algorithm (jointly with Sagi Snir) to construct a phylogenetic tree by computing only n SVD’s. Our algorithm only assumes that evolution follows a Markov model on a binary tree and that evolutions happens independently at different sites of the genome. No assumptions are made about the shape of the transition matrices. The algorithm uses phylogenetic invariants. Such polynomials have been studied for years (e.g., [1, 2]) and have been used to infer phylogenies on four and five taxa ([7]), but have been widely considered impractical. However, our algorithm is very fast in practice on trees with up to 15-25 taxa. It shows promise for real data because it does not assume the existence of a global rate matrix; for example, it places the rodents correctly more often than other methods do. We have implemented this algorithm using the SVDLIBC library ([6]) and have done extensive testing with simulated and real data.