  In this paper, we propose a unified algorithmic framework for solving many
known variants of \mds. Our algorithm is a simple iterative scheme with
guaranteed convergence, and is \emph{modular}; by changing the internals of a
single subroutine in the algorithm, we can switch cost functions and target
spaces easily. In addition to the formal guarantees of convergence, our
algorithms are accurate; in most cases, they converge to better quality
solutions than existing methods, in comparable time. We expect that this
framework will be useful for a number of \mds variants that have not yet been
studied.
  Our framework extends to embedding high-dimensional points lying on a sphere
to points on a lower dimensional sphere, preserving geodesic distances. As a
compliment to this result, we also extend the Johnson-Lindenstrauss Lemma to
this spherical setting, where projecting to a random $O((1/\eps^2) \log
n)$-dimensional sphere causes $\eps$-distortion.
