Incoherent Dictionary Learning Method with Border Condition Constrained for Sparse Representation
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摘要: 从字典的相干性边界条件出发, 提出一种基于极分解的非相干字典学习方法(Polar decomposition based incoherent dictionary learning, PDIDL), 该方法将字典以Frobenius范数逼近由矩阵极分解获取的紧框架, 同时采用最小化所有原子对的内积平方和作为约束, 以降低字典的相干性, 并保持更新前后字典结构的整体相似特性. 采用最速梯度下降法和子空间旋转实现非相干字典的学习和优化. 最后将该方法应用于合成数据与实际语音数据的稀疏表示. 实验结果表明, 本文方法学习的字典能逼近等角紧框架(Equiangular tight-frame, ETF), 实现最大化稀疏编码, 在降低字典相干性的同时具有较低的稀疏表示误差.Abstract: A novel incoherent dictionary learning method is introduced based on polar decomposition, which is simple but practical framework for designing incoherent dictionary. The method distinguishes itself from traditional dictionary learning approaches by explicitly taking into account the border condition. In particular, the dictionary approximates the r-tight frame obtained with matrix polar decomposition with respect to Frobenius norm. The minimization of the sum of squared inner product of all atom pairs is taken as an constraint so as to reduce the dictionary coherence and keep the overall dictionary structure similarity before and after the update. Incoherent dictionary learning and optimization are performed based on the steepest gradient descent method and subspace rotation. Experiments on synthetic data and real audio data show that the proposed approach can achieve considerable improvements in trems of approximating equiangular tight frame (ETF) and implementing the maximal sparse coding, and can effective control the trade-off between low sparse representation errors and dictionary coherence in comparison to other methods.
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