10,158 research outputs found
Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe
(FW) algorithms regained popularity in recent years due to their simplicity,
effectiveness and theoretical guarantees. MP and FW address optimization over
the linear span and the convex hull of a set of atoms, respectively. In this
paper, we consider the intermediate case of optimization over the convex cone,
parametrized as the conic hull of a generic atom set, leading to the first
principled definitions of non-negative MP algorithms for which we give explicit
convergence rates and demonstrate excellent empirical performance. In
particular, we derive sublinear () convergence on general
smooth and convex objectives, and linear convergence () on
strongly convex objectives, in both cases for general sets of atoms.
Furthermore, we establish a clear correspondence of our algorithms to known
algorithms from the MP and FW literature. Our novel algorithms and analyses
target general atom sets and general objective functions, and hence are
directly applicable to a large variety of learning settings.Comment: NIPS 201
Robust computation of linear models by convex relaxation
Consider a dataset of vector-valued observations that consists of noisy
inliers, which are explained well by a low-dimensional subspace, along with
some number of outliers. This work describes a convex optimization problem,
called REAPER, that can reliably fit a low-dimensional model to this type of
data. This approach parameterizes linear subspaces using orthogonal projectors,
and it uses a relaxation of the set of orthogonal projectors to reach the
convex formulation. The paper provides an efficient algorithm for solving the
REAPER problem, and it documents numerical experiments which confirm that
REAPER can dependably find linear structure in synthetic and natural data. In
addition, when the inliers lie near a low-dimensional subspace, there is a
rigorous theory that describes when REAPER can approximate this subspace.Comment: Formerly titled "Robust computation of linear models, or How to find
a needle in a haystack
- …