We analyze alternating descent algorithms for minimizing the sum of a quadratic function and block separable non-smooth functions. In case the quadratic interactions between the blocks are pairwise, we show that the schemes can be accelerated, leading to improved convergence rates with respect to related accelerated parallel proximal descent. As an application we obtain very fast algorithms for computing the proximity operator of the 2D and 3D total variation.

Keywords: block coordinate descent, Dykstra’s algorithms, first order methods, acceleration, FISTA

Antonin Chambolle ^{1};
Thomas Pock ^{2}

@article{SMAI-JCM_2015__1__29_0, author = {Antonin Chambolle and Thomas Pock}, title = {A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions}, journal = {The SMAI Journal of computational mathematics}, pages = {29--54}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {1}, year = {2015}, doi = {10.5802/smai-jcm.3}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.3/} }

TY - JOUR AU - Antonin Chambolle AU - Thomas Pock TI - A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions JO - The SMAI Journal of computational mathematics PY - 2015 SP - 29 EP - 54 VL - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.3/ DO - 10.5802/smai-jcm.3 LA - en ID - SMAI-JCM_2015__1__29_0 ER -

%0 Journal Article %A Antonin Chambolle %A Thomas Pock %T A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions %J The SMAI Journal of computational mathematics %D 2015 %P 29-54 %V 1 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.3/ %R 10.5802/smai-jcm.3 %G en %F SMAI-JCM_2015__1__29_0

Antonin Chambolle; Thomas Pock. A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions. The SMAI Journal of computational mathematics, Volume 1 (2015), pp. 29-54. doi : 10.5802/smai-jcm.3. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.3/

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