  We introduce new methods for phylogenetic tree quartet construction by using
machine learning to optimize the power of phylogenetic invariants. Phylogenetic
invariants are polynomials in the joint probabilities which vanish under a
model of evolution on a phylogenetic tree. We give algorithms for selecting a
good set of invariants and for learning a metric on this set of invariants
which optimally distinguishes the different models. Our learning algorithms
involve linear and semidefinite programming on data simulated over a wide range
of parameters. We provide extensive tests of the learned metrics on simulated
data from phylogenetic trees with four leaves under the Jukes-Cantor and Kimura
3-parameter models of DNA evolution. Our method greatly improves on other uses
of invariants and is competitive with or better than neighbor-joining. In
particular, we obtain metrics trained on trees with short internal branches
which perform much better than neighbor joining on this region of parameter
space.
