A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group. Following this analogy, we can define relevant and irrelevant operators, where the role of dimensionality is played by properties of the eigenvalue density. These results also suggest an approach to the analysis of real data. As an example, we study neural activity in the vertebrate retina as it responds to naturalistic movies, and find evidence of behavior controlled by a nontrivial fixed point. Applied to financial data, our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non-Gaussian distribution.

中文翻译：

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PCA 遇到 RG

具有多个自由度的系统可以用协方差矩阵来表征；主成分分析主要针对这个矩阵的特征值，希望能找到一个更低维的描述。但是当频谱几乎是连续的时，我们保留的组件和我们忽略的组件之间的任何区别都变得随意；那么很自然地会问，当我们改变这个任意截止值时会发生什么。我们认为这个问题类似于动量壳重整化群。按照这个类比，我们可以定义相关和不相关的算子，其中维数的作用由特征值密度的属性发挥。这些结果还提出了一种分析真实数据的方法。例如，我们研究脊椎动物视网膜对自然电影的反应时的神经活动，并找到由非平凡不动点控制的行为的证据。应用于金融数据，我们的分析将采样噪声主导的模式与非高斯分布描述的较小但仍然宏观数量的模式分开。