Randomized Sparse Block Kaczmarz as Randomized Dual Block-Coordinate Descent

• Published in: Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică Vol. 23; no. 3; pp. 129 - 149
• Main Author: Petra, Stefania
• Format: Journal Article
• Language: English
• Published: Constanta De Gruyter Open 01.11.2015; De Gruyter Brill Sp. z o.o., Paradigm Publishing Services; Sciendo
• Subjects: compressed sensing; convex duality; convex programming; limited-data tomography; linearized Bregman method; Randomized block-coordinate descent; sparse Kaczmarz solver; underdetermined system of linear equations
• ISSN: 1844-0835; 1224-1784; 1844-0835
• DOI: 10.1515/auom-2015-0052

We show that the Sparse Kaczmarz method is a particular instance of the coordinate gradient method applied to an unconstrained dual problem corresponding to a regularized ℓ -minimization problem subject to linear constraints. Based on this observation and recent theoretical work concerning the convergence analysis and corresponding convergence rates for the randomized block coordinate gradient descent method, we derive block versions and consider randomized ordering of blocks of equations. Convergence in expectation is thus obtained as a byproduct. By smoothing the ℓ -objective we obtain a strongly convex dual which opens the way to various acceleration schemes.