标题： 拉姆齐理论揭示了在子采样数据上稀疏编码的唯一性条件 显示英文标题

标题： Ramsey theory reveals the conditions when sparse coding on subsampled data is unique

摘要： 稀疏编码或字典学习已被广泛用于揭示多种感觉数据中的稀疏底层结构。 信号处理中的一项相关进展是压缩感知，这是一种解释如何将稀疏数据以低于奈奎斯特-香农极限的速率进行采样，并从这些采样中高效恢复的理论。 在这里，我们研究压缩感知中的恢复条件是否足以使字典学习发现采样数据的原始稀疏原因。 利用组合Ramsey理论，我们完全表征了何时学习得到的字典矩阵和采样数据的稀疏表示是唯一的（在排列和缩放的自然等价性下）。 令人惊讶的是，无需对学习得到的字典或推断出的稀疏代码做出任何假设，唯一性就被证明成立。 我们的结果对于从采样数据中学习过完备字典具有重要意义，并在数据分析和神经科学中具有潜在应用。 例如，它将稀疏编码识别为通过严重瓶颈实现无损通信的一种可能学习机制，这可能解释不同脑区如何通过轴突纤维投射进行通信。 显示英文摘要

摘要： Sparse coding or dictionary learning has been widely used to reveal the sparse underlying structure of many kinds of sensory data. A related advance in signal processing is compressed sensing, a theory explaining how sparse data can be subsampled below the Nyquist-Shannon limit and then efficiently recovered from these subsamples. Here we study whether the conditions for recovery in compressed sensing are sufficient for dictionary learning to discover the original sparse causes of subsampled data. Using combinatorial Ramsey theory, we completely characterize when the learned dictionary matrix and sparse representations of subsampled data are unique (up to the natural equivalences of permutation and scaling). Surprisingly, uniqueness is shown to hold without any assumptions on the learned dictionaries or inferred sparse codes. Our result has implications for the learning of overcomplete dictionaries from subsampled data and has potential applications in data analysis and neuroscience. For instance, it identifies sparse coding as a possible learning mechanism for establishing lossless communication through severe bottlenecks, which might explain how different brain regions communicate through axonal fiber projections.