443 research outputs found
Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold
The conjugate gradient method is a crucial first-order optimization method
that generally converges faster than the steepest descent method, and its
computational cost is much lower than the second-order methods. However, while
various types of conjugate gradient methods have been studied in Euclidean
spaces and on Riemannian manifolds, there has little study for those in
distributed scenarios. This paper proposes a decentralized Riemannian conjugate
gradient descent (DRCGD) method that aims at minimizing a global function over
the Stiefel manifold. The optimization problem is distributed among a network
of agents, where each agent is associated with a local function, and
communication between agents occurs over an undirected connected graph. Since
the Stiefel manifold is a non-convex set, a global function is represented as a
finite sum of possibly non-convex (but smooth) local functions. The proposed
method is free from expensive Riemannian geometric operations such as
retractions, exponential maps, and vector transports, thereby reducing the
computational complexity required by each agent. To the best of our knowledge,
DRCGD is the first decentralized Riemannian conjugate gradient algorithm to
achieve global convergence over the Stiefel manifold
A new, globally convergent Riemannian conjugate gradient method
This article deals with the conjugate gradient method on a Riemannian
manifold with interest in global convergence analysis. The existing conjugate
gradient algorithms on a manifold endowed with a vector transport need the
assumption that the vector transport does not increase the norm of tangent
vectors, in order to confirm that generated sequences have a global convergence
property. In this article, the notion of a scaled vector transport is
introduced to improve the algorithm so that the generated sequences may have a
global convergence property under a relaxed assumption. In the proposed
algorithm, the transported vector is rescaled in case its norm has increased
during the transport. The global convergence is theoretically proved and
numerically observed with examples. In fact, numerical experiments show that
there exist minimization problems for which the existing algorithm generates
divergent sequences, but the proposed algorithm generates convergent sequences.Comment: 22 pages, 8 figure
Blind Source Separation with Compressively Sensed Linear Mixtures
This work studies the problem of simultaneously separating and reconstructing
signals from compressively sensed linear mixtures. We assume that all source
signals share a common sparse representation basis. The approach combines
classical Compressive Sensing (CS) theory with a linear mixing model. It allows
the mixtures to be sampled independently of each other. If samples are acquired
in the time domain, this means that the sensors need not be synchronized. Since
Blind Source Separation (BSS) from a linear mixture is only possible up to
permutation and scaling, factoring out these ambiguities leads to a
minimization problem on the so-called oblique manifold. We develop a geometric
conjugate subgradient method that scales to large systems for solving the
problem. Numerical results demonstrate the promising performance of the
proposed algorithm compared to several state of the art methods.Comment: 9 pages, 2 figure
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