标题： 关于卷积稀疏字典学习的重建风险 显示英文标题

标题： On the Reconstruction Risk of Convolutional Sparse Dictionary Learning

摘要： 稀疏字典学习（SDL）已成为一种流行的自适应方法，用于识别数据集的简洁表示，这是机器学习和信号处理中的一个基本问题。尽管大多数关于SDL的工作都假设训练数据集是由独立同分布样本组成，但一种称为卷积稀疏字典学习（CSDL）的变体放松了这一假设，允许更一般的顺序数据源，例如时间序列或其他相关数据。 虽然最近的研究已经探索了经典SDL的统计性质，但CSDL的统计性质仍未被研究。 本文通过确定CSDL在重建风险下的最小最大收敛率开始研究这个问题，该方法既包括对已建立的CSDL估计器的风险上界，也包括证明一个匹配的信息论下界。 我们的结果表明，精确在“超稀疏”设置下可以实现重建风险的一致性，即稀疏度（即特征出现次数）与训练序列长度N成$o(N)$关系。 值得注意的是，我们的结果做出了非常弱的假设，允许任意字典和相关的测量噪声。 最后，我们用合成数据上的数值实验验证了我们的理论结果。 显示英文摘要

摘要： Sparse dictionary learning (SDL) has become a popular method for adaptively identifying parsimonious representations of a dataset, a fundamental problem in machine learning and signal processing. While most work on SDL assumes a training dataset of independent and identically distributed samples, a variant known as convolutional sparse dictionary learning (CSDL) relaxes this assumption, allowing more general sequential data sources, such as time series or other dependent data. Although recent work has explored the statistical properties of classical SDL, the statistical properties of CSDL remain unstudied. This paper begins to study this by identifying the minimax convergence rate of CSDL in terms of reconstruction risk, by both upper bounding the risk of an established CSDL estimator and proving a matching information-theoretic lower bound. Our results indicate that consistency in reconstruction risk is possible precisely in the `ultra-sparse' setting, in which the sparsity (i.e., the number of feature occurrences) is in $o(N)$ in terms of the length N of the training sequence. Notably, our results make very weak assumptions, allowing arbitrary dictionaries and dependent measurement noise. Finally, we verify our theoretical results with numerical experiments on synthetic data.