l0 Sparsifying Transform Learning With Efficient Optimal Updates and Convergence Guarantees
l0 Sparsifying Transform Learning With Efficient Optimal Updates and Convergence Guarantees
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DOI：
10.1109/tsp.2015.2405503
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发表时间：
2015-05-01
影响因子：
5.4
通讯作者：
Bresler, Yoram
中科院分区：
文献类型：
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作者：
Ravishankar, Saiprasad;Bresler, Yoram
信号处理中的许多应用得益于信号在某个变换域或字典中的稀疏性。合成稀疏字典直接适应于数据，在图像去噪、修复和医学图像重建等应用中得到了广泛应用。在本文中，我们将重点放在稀疏变换模型上，并研究条件良好的正方形稀疏变换的学习。所提出的算法在基于“范数”的稀疏编码步骤和非凸变换更新步骤之间交替。我们给出了每一步的精确解析解。提出的变换更新步骤的解决方案在该步骤中达到全局最小值，并且还提供了与涉及共轭梯度的迭代解相比的加速比。我们证明了我们的交替算法全局收敛于非凸变换学习问题的局部极小值集。在实际应用中，这些算法对初始化不敏感。实验结果表明，变换学习比合成K-SVD在图像去噪中具有良好的性能和显著的加速效果。
Many applications in signal processing benefit from the sparsity of signals in a certain transform domain or dictionary. Synthesis sparsifying dictionaries that are directly adapted to data have been popular in applications such as image denoising, inpainting, and medical image reconstruction. In this paper, we focus instead on the sparsifying transform model, and study the learning of well-conditioned square sparsifying transforms. The proposed algorithms alternate between a "norm"-based sparse coding step, and a non-convex transform update step. We derive the exact analytical solution for each of these steps. The proposed solution for the transform update step achieves the global minimum in that step, and also provides speedups over iterative solutions involving conjugate gradients. We establish that our alternating algorithms are globally convergent to the set of local minimizers of the nonconvex transform learning problems. In practice, the algorithms are insensitive to initialization. We present results illustrating the promising performance and significant speed-ups of transform learning over synthesis K-SVD in image denoising.