  An alternative to independent pairwise distance estimation is proposed that
uses hyperbolic geometry to jointly estimate pairwise distances subject to a
weakening of the four point condition that characterises tree metrics.
Specifically, taxa are represented as points in hyperbolic space such that the
distance between a pair of points accounts for the site differences between the
corresponding taxa. The proposed algorithm iteratively rearranges the points to
increase an objective function that is shown empirically to increase the
log-likelihood employed in tree search. Unlike the log-likelihood on tree
space, the proposed objective function is differentiable, allowing for the use
of gradient-based techniques in its optimisation. It is shown that the error
term in the weakened four point condition is bounded by a linear function of
the radius parameter of the hyperboloid model, which controls the curvature of
the space. The error may thus be made as small as desired, within the bounds of
computational precision.
