165 research outputs found
A Riemannian Primal-dual Algorithm Based on Proximal Operator and its Application in Metric Learning
In this paper, we consider optimizing a smooth, convex, lower semicontinuous
function in Riemannian space with constraints. To solve the problem, we first
convert it to a dual problem and then propose a general primal-dual algorithm
to optimize the primal and dual variables iteratively. In each optimization
iteration, we employ a proximal operator to search optimal solution in the
primal space. We prove convergence of the proposed algorithm and show its
non-asymptotic convergence rate. By utilizing the proposed primal-dual
optimization technique, we propose a novel metric learning algorithm which
learns an optimal feature transformation matrix in the Riemannian space of
positive definite matrices. Preliminary experimental results on an optimal fund
selection problem in fund of funds (FOF) management for quantitative investment
showed its efficacy.Comment: 8 pages, 2 figures, published as a conference paper in 2019
International Joint Conference on Neural Networks (IJCNN
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Mini-Workshop: Computational Optimization on Manifolds (online meeting)
The goal of the mini-workshop was to study the geometry, algorithms and applications of unconstrained and constrained optimization problems posed on Riemannian manifolds.
Focus topics included the geometry of particular manifolds, the formulation and analysis of a number of application problems, as well as novel algorithms and their implementation
Extragradient Type Methods for Riemannian Variational Inequality Problems
Riemannian convex optimization and minimax optimization have recently drawn
considerable attention. Their appeal lies in their capacity to adeptly manage
the non-convexity of the objective function as well as constraints inherent in
the feasible set in the Euclidean sense. In this work, we delve into monotone
Riemannian Variational Inequality Problems (RVIPs), which encompass both
Riemannian convex optimization and minimax optimization as particular cases. In
the context of Euclidean space, it is established that the last-iterates of
both the extragradient (EG) and past extragradient (PEG) methods converge to
the solution of monotone variational inequality problems at a rate of
(Cai et al., 2022). However, analogous
behavior on Riemannian manifolds remains an open question. To bridge this gap,
we introduce the Riemannian extragradient (REG) and Riemannian past
extragradient (RPEG) methods. We demonstrate that both exhibit
last-iterate convergence. Additionally, we
show that the average-iterate convergence of both REG and RPEG is
, aligning with observations in the Euclidean case
(Mokhtari et al., 2020). These results are enabled by judiciously addressing
the holonomy effect so that additional complications in Riemannian cases can be
reduced and the Euclidean proof inspired by the performance estimation problem
(PEP) technique or the sum-of-squares (SOS) technique can be applied again
Strong Convexity of Sets in Riemannian Manifolds
Convex curvature properties are important in designing and analyzing convex
optimization algorithms in the Hilbertian or Riemannian settings. In the case
of the Hilbertian setting, strongly convex sets are well studied. Herein, we
propose various definitions of strong convexity for uniquely geodesic sets in a
Riemannian manifold. We study their relationship, propose tools to determine
the geodesic strongly convex nature of sets, and analyze the convergence of
optimization algorithms over those sets. In particular, we demonstrate that the
Riemannian Frank-Wolfe algorithm enjoys a global linear convergence rate when
the Riemannian scaling inequalities hold
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer
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