Nonnegative Dictionary-Learning Algorithm Based on L1 Norm with the Sparse Analysis Model
Abstract
Sparse representation has been proven to be a powerful tool for analysis and processing of signals and images. Most of existing methods for sparse representation are based on the synthesis model. This paper presents a method for dictionary learning and sparse representation with the so-called analysis model. Different from the synthesis sparse model, in this analysis model, the analysis dictionary multiplying the signal can lead to a sparse outcome. The analysis dictionary learning problem has received less attention with and only a few algorithms has been proposed recently. What is more, there have still been few investigations in the context of dictionary learning for nonnegative signal representation. So, in this paper we focus on the nonnegative dictionary learning for signal representation. We use ℓ1-norm as the sparsity measure to learn an analysis dictionary from signals in analysis sparse model. In addition, we adopt the Euclidean distance as the error measure in the formulation. Numerical experiments on recovery of analysis dictionary show that the proposed analysis dictionary learning algorithm performs well for nonnegative signal representation.
References
R. Rubinstein, T. Peleg, and M. Elad, Analysis K-SVD: A dictionary-learning algorithm for the analysis sparse model, IEEE Trans. on Signal Processing, vol. 61, no. 3, pp.661-677, 2013.
Z. Tang, S. Ding, Z. Yang, Dictionary learning for sparse representation by nonnegative matrix factorization with constraint of determinant-type of maximization, ICIC Express Letter. vol. 4, no. 5, October 2010.
S. Nam, M. E. Davies, M. Elad, and R. Gribonval, The cosparse analysis model and algorithms, Appl. Comput. Harmon. Anal., vol. 34, no. 1, pp. 30-56, 2013.
M. Elad, Sparse and Redundant Representations From Theory to Applications in Signal and Image Processing. NewYork: Springer, 2010.
J. Starck, F. Murtagh, and M. Fadili, Sparse Image and Signal Processing Wavelets, Curvelets, Morphological Diversity. Cambridge, U.K.: Cambridge Univ. Press, 2010.
Y. Li, S. Ding, and Z. Li, A dictionary-learning algorithm for the analysis sparse model with a determinant-type of sparsity measure, in IEEE International Conference on Digital Signal Processing (DSP), 2014, pp. 152-156.
S. Ravishankar, and Y. Bresler, Learning Sparsifying Transforms, IEEE Trans. on Signal Processing, vol. 61, no. 5, pp. 1072-1086, 2013.
R. Giryesa, S. Namb, M. Elada, R.Gribonvalb, and M. E. Daviesc, Greedy-Like Algorithms for the Cosparse Analysis Model, Linear Alg. Appl., 2013, [online] Available: http://www.sciencedirect.com/science/article/pii/S0024379513001870.
T. Peleg and M. Elad, Performance guarantees of the thresholding algorithm for the cosparse analysis model, IEEE Trans. Inf. Theory., Vol. 59, no. 3, pp. 1832-1845, 2013.
B. Ophir, M. Elad, N. Bertin, and M. D. Plumbley, Sequential minimal eigenvalues: an approach to analysis dictionary learning, in Proceedings of the 9th European Signal Processing Conference, 2011, pp. 1465-1469.
M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, Constrained overcomplete analysis operator learning for cosparse signal modelling, IEEE Trans. on Signal Processing, vol. 61, no. 9, pp. 2341-2355, 2013.
Y. Zhang, H. Wang, T. Yu, and W. Wang, Subset pursuit for analysis dictionary learning, in Signal Processing Conference (EUSIPCO), 2013, pp. 1-5.
Y. Zhang, W. Wang, H. Wang, and S. Sanei, Kplane clustering algorithmfor analysis dictionary learning, in Proc. of the IEEE International Workshop on Machine Learning for Signal Processing, 2013, pp. 1-4.
J. M. Duarte-Carvajalino and G. Sapiro, Learning to Sense Sparse Signals: Simultaneous Sensing Matrix and Sparsifying Dictionary Optimization, IEEE Trans. on Image Processing, vol. 18, no. 7, pp. 1395-1408, 2009.
Cedric A. Zala, Ian Barrodale, and J. S. Kennedy, High-Resolution Signal and Noise Field Estimation Using the L1 (Least Absolute Values) Norm, IEEE Journal of Oceanic Engineering, Vol. OE-12, no. 1, January 1987.
P. O. Hoyer, Non-negative matrix factorization with sparseness constraints, J. Mach. Learn. Res.,vol. 5, no. 37, pp. 1457-1469, 2004.
Y. Zhang, H. Wang, T. Yu, and W. Wang, Subset pursuit for analysis dictionary learning, in Signal Processing Conference (EUSIPCO), 2013, pp. 1-5.
