A simple and efficient algorithm for fused lasso signal approximator with convex loss function
 Lichun Wang,
 Yuan You,
 Heng Lian
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We consider the augmented Lagrangian method (ALM) as a solver for the fused lasso signal approximator (FLSA) problem. The ALM is a dual method in which squares of the constraint functions are added as penalties to the Lagrangian. In order to apply this method to FLSA, two types of auxiliary variables are introduced to transform the original unconstrained minimization problem into a linearly constrained minimization problem. Each updating in this iterative algorithm consists of just a simple onedimensional convex programming problem, with closed form solution in many cases. While the existing literature mostly focused on the quadratic loss function, our algorithm can be easily implemented for general convex loss. We also provide some convergence analysis of the algorithm. Finally, the method is illustrated with some simulation datasets.
 Conte, SD, Boor, C (1980) Elementary numerical analysis : an algorithmic approach. McGrawHill, New York
 Ekeland, I, Turnbull, T (1983) Infinitedimensional optimization and convexity. Chicago lectures in mathematics. University of Chicago Press, Chicago
 Friedman, J, Hastie, T, Hofling, H, Tibshirani, R (2007) Pathwise coordinate optimization. Ann Appl Stat 1: pp. 302332 CrossRef
 Glowinski, R, Tallec, P (1989) Augmented Lagrangian and operatorsplitting methods in nonlinear mechanics. Society for Industrial and Applied Mathematics, Philadelphia CrossRef
 Hestenes, MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4: pp. 303320 CrossRef
 Hoefling H (2010) A path algorithm for the fused lasso signal approximator. J Comput Graph Stat 19(4): 984–1006
 Huang, T, Wu, BL, Lizardi, P, Zhao, HY (2005) Detection of DNA copy number alterations using penalized least squares regression. Bioinformatics 21: pp. 38113817 CrossRef
 Powell MJD (1969) A method for nonlinear constraints in minimization problems. In: Fletcher R (ed) Optimization. Academic Press, New York, pp 283–298
 Rockafellar, RT (1970) Convex analysis. Princeton University Press, Princeton
 Rosset, S, Zhu, J (2007) Piecewise linear regularized solution paths. Ann Stat 35: pp. 10121030 CrossRef
 Tai XC, Wu C (2009) Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In: 2nd international conference on scale space and variational methods in computer vision. pp 502–513
 Tao, M, Yuan, X (2011) Recovering lowrank and sparse components of matrices from incomplete and noisy observations. SIAM J Optim 21: pp. 5781 CrossRef
 Tibshirani, R, Saunders, M, Rosset, S, Zhu, J, Knight, K (2005) Sparsity and smoothness via the fused lasso. J R Stat Soc Ser BStat Methodology 67: pp. 91108 CrossRef
 Tibshirani, R, Wang, P (2008) Spatial smoothing and hot spot detection for CGH data using the fused lasso. Biostatistics 9: pp. 1829 CrossRef
 Wen, Z, Goldfarb, D, Yin, W (2010) Alternating direction augmented Lagrangian methods for semidefinite programming. Math Program Comput 2: pp. 203230 CrossRef
 Yang J, Yuan X (2012) Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization. Math Comput 82:301–329
 Yang J, Zhang Y (2011) Alternating direction algorithms for \(l_{1}\) problems in compressive sensing. SIAM J Scientific Comput 33(1):250–278
 Zou, H, Li, RZ (2008) Onestep sparse estimates in nonconcave penalized likelihood models. Ann Stat 36: pp. 15091533 CrossRef
 Title
 A simple and efficient algorithm for fused lasso signal approximator with convex loss function
 Journal

Computational Statistics
Volume 28, Issue 4 , pp 16991714
 Cover Date
 20130801
 DOI
 10.1007/s0018001203736
 Print ISSN
 09434062
 Online ISSN
 16139658
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Augmented Lagrangian
 Convergence analysis
 LADFLASSO
 Industry Sectors
 Authors

 Lichun Wang ^{(1)}
 Yuan You ^{(2)}
 Heng Lian ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, PR China
 2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore