1,956 research outputs found
PRISMA: PRoximal Iterative SMoothing Algorithm
Motivated by learning problems including max-norm regularized matrix
completion and clustering, robust PCA and sparse inverse covariance selection,
we propose a novel optimization algorithm for minimizing a convex objective
which decomposes into three parts: a smooth part, a simple non-smooth Lipschitz
part, and a simple non-smooth non-Lipschitz part. We use a time variant
smoothing strategy that allows us to obtain a guarantee that does not depend on
knowing in advance the total number of iterations nor a bound on the domain
A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints
We propose a new algorithm to solve optimization problems of the form for a smooth function under the constraints that is positive
semidefinite and the diagonal blocks of are small identity matrices. Such
problems often arise as the result of relaxing a rank constraint (lifting). In
particular, many estimation tasks involving phases, rotations, orthonormal
bases or permutations fit in this framework, and so do certain relaxations of
combinatorial problems such as Max-Cut. The proposed algorithm exploits the
facts that (1) such formulations admit low-rank solutions, and (2) their
rank-restricted versions are smooth optimization problems on a Riemannian
manifold. Combining insights from both the Riemannian and the convex geometries
of the problem, we characterize when second-order critical points of the smooth
problem reveal KKT points of the semidefinite problem. We compare against state
of the art, mature software and find that, on certain interesting problem
instances, what we call the staircase method is orders of magnitude faster, is
more accurate and scales better. Code is available.Comment: 37 pages, 3 figure
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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