On the Convergence of Block Majorization-Minimization Algorithms on the Grassmann Manifold

The Majorization-Minimization (MM) framework is widely used to derive efficient algorithms for specific problems that require the optimization of a cost function (which can be convex or not). It is based on a sequential optimization of a surrogate function over closed convex sets. A natural extensio...

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Bibliographic Details
Published in:IEEE signal processing letters Vol. 31; pp. 1314 - 1318
Main Authors: Lopez, Carlos Alejandro, Riba, Jaume
Format: Journal Article
Language:English
Published: New York IEEE 2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Constraints
Convergence
Convexity
Cost function
geodesically convex optimization
Grassmann manifold
Independent variables
majorization-minimization
Manifolds
Manifolds (mathematics)
Minimization
Non-convex optimization
Optimization
Principal component analysis
Riemannian optimization
Signal processing
Signal processing algorithms
ISSN:1070-9908, 1558-2361
Online Access:Get full text
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Summary:The Majorization-Minimization (MM) framework is widely used to derive efficient algorithms for specific problems that require the optimization of a cost function (which can be convex or not). It is based on a sequential optimization of a surrogate function over closed convex sets. A natural extension of this framework incorporates ideas of Block Coordinate Descent (BCD) algorithms into the MM framework, also known as block MM. The rationale behind the block extension is to partition the optimization variables into several independent blocks, to obtain a surrogate for each block, and to optimize the surrogate of each block cyclically. However, known convergence proofs of the block MM are only valid under the assumption that the constraint sets are closed and convex. Hence, the global convergence of the block MM is not ensured for non-convex sets by classical proofs, which is needed in iterative schemes that naturally emerge in a wide range of subspace-based signal processing applications. For this purpose, the aim of this letter is to review the convergence proof of the block MM and extend it for blocks constrained in the Grassmann manifold.
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ISSN:1070-9908
1558-2361
DOI:10.1109/LSP.2024.3396660