标题： 在什么情况下字典学习可以从子采样中唯一地恢复稀疏数据？ 显示英文标题

标题： When can dictionary learning uniquely recover sparse data from subsamples?

摘要： 稀疏编码或稀疏字典学习已被广泛用于恢复各种自然数据中的潜在结构。 在这里，我们提供了保证这种恢复普遍性的条件；即，当稀疏编码和字典是唯一的（在自然对称性下）。 我们的主要工具是组合矩阵理论中的一个有用引理，它使我们能够在各种训练数据生成假设下推导出保证这种唯一性的样本大小的界限。 每当我们的定理中的条件得到满足时，任何能够重建数据的稀疏性约束学习算法都会恢复原始的稀疏编码和字典。 我们还讨论了其在神经科学和数据分析中的潜在应用。 显示英文摘要

摘要： Sparse coding or sparse dictionary learning has been widely used to recover underlying structure in many kinds of natural data. Here, we provide conditions guaranteeing when this recovery is universal; that is, when sparse codes and dictionaries are unique (up to natural symmetries). Our main tool is a useful lemma in combinatorial matrix theory that allows us to derive bounds on the sample sizes guaranteeing such uniqueness under various assumptions for how training data are generated. Whenever the conditions to one of our theorems are met, any sparsity-constrained learning algorithm that succeeds in reconstructing the data recovers the original sparse codes and dictionary. We also discuss potential applications to neuroscience and data analysis.