Abstract
We introduce the Group Total Variation (GTV) regularizer, a modification of Total Variation that uses the ℓ2,1 norm instead of the ℓ1 one to deal with multidimensional features. When used as the only regularizer, GTV can be applied jointly with iterative convex optimization algorithms such as FISTA. This requires to compute its proximal operator which we derive using a dual formulation. GTV can also be combined with a Group Lasso (GL) regularizer, leading to what we call Group Fused Lasso (GFL) whose proximal operator can now be computed combining the GTV and GL proximals through Dykstra algorithm. We will illustrate how to apply GFL in strongly structured but ill-posed regression problems as well as the use of GTV to denoise colour images.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Bach, F., Jenatton, R., Mairal, J., Obozinski, G.: Convex Optimization with Sparsity-Inducing Norms (2011), http://www.di.ens.fr/~fbach/opt_book.pdf
Barbero, A., Sra, S.: Fast newton–type methods for total variation regularization. In: Proceedings of the 28th International Conference on Machine Learning (ICML 2011), New York, NY, USA, pp. 313–320 (2011)
Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1), 183–202 (2009)
Bioucas-Dias, J.M., Figueiredo, M.A.T.: A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Transactions on Image Processing 16(12), 2992–3004 (2007)
Bleakley, K., Vert, J.P.: The group fused Lasso for multiple change-point detection. ArXiv e-prints (2011)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. Recherche 49, 1–25 (2009)
Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58(1), 267–288 (1996)
Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., Knight, K.: Sparsity and smoothness via the fused lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(1), 91–108 (2005)
Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society – Series B: Statistical Methodology 68(1), 49–67 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alaíz, C.M., Barbero, Á., Dorronsoro, J.R. (2013). Group Fused Lasso. In: Mladenov, V., Koprinkova-Hristova, P., Palm, G., Villa, A.E.P., Appollini, B., Kasabov, N. (eds) Artificial Neural Networks and Machine Learning – ICANN 2013. ICANN 2013. Lecture Notes in Computer Science, vol 8131. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40728-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-40728-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40727-7
Online ISBN: 978-3-642-40728-4
eBook Packages: Computer ScienceComputer Science (R0)