Minimax Reconstruction Risk of Convolutional Sparse Dictionary Learning
Minimax Reconstruction Risk of Convolutional Sparse Dictionary Learning
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发表时间：
2018-03
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通讯作者：
Shashank Singh;B. Póczos;Jian Ma
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作者：
Shashank Singh;B. Póczos;Jian Ma
稀疏字典学习（SDL）已经成为学习数据精简表示的一种流行方法，这是机器学习和信号处理中的一个基本问题。虽然大多数关于SDL的工作假设一个独立且同分布（IID）样本的训练数据集，但一种称为卷积稀疏字典学习（CSDL）的变体放宽了这一假设，以允许依赖的、非平稳的顺序数据源。最近的工作探索了IID SDL的统计特性；然而，CSDL的统计特性在很大程度上仍未得到研究。本文从重建风险的角度确定了CSDL的最小最大值，并提供了各种情况下的下限和上限。我们的结果做了最小的假设，允许任意字典，并表明CSDL对相关噪声具有鲁棒性。我们将我们的结果与IID SDL的类似结果进行了比较，并用合成实验验证了我们的理论。
Sparse dictionary learning (SDL) has become a popular method for learning parsimonious representations of data, a fundamental problem in machine learning and signal processing. While most work on SDL assumes a training dataset of independent and identically distributed (IID) samples, a variant known as convolutional sparse dictionary learning (CSDL) relaxes this assumption to allowdependent, non-stationary sequential data sources. Recent work has explored statistical properties of IID SDL; however, the statistical properties of CSDL remain largely unstudied. This paper identifies minimax rates of CSDL in terms of reconstruction risk, providing both lower and upper bounds in a variety of settings. Our results make minimal assumptions, allowing arbitrary dictionaries and showing that CSDL is robust to dependent noise. We compare our results to similar results for IID SDL and verify our theory with synthetic experiments.