AI 中文总结
本文证明PCA在测量尺度连续变化下，各主成分按特征值排序的阶次保持不变，即模式具有“阶稳定性”，并解释了正交缩放时特征值交叉现象实为模式瞬时交换方向。
AI 中文摘要
PCA算法对测量尺度的变化敏感。例如，将系统中某个变量的测量单位从厘米改为英寸，会改变其主轴和主特征值。尽管这种尺度依赖性通常很复杂，但我们在此证明它仍然遵循一个严格的不变性性质：在连续的尺度调整下，初始状态的第k大主成分（按特征值排序）连续演变为最终状态的第k大主成分，对于每个k都成立。从这个意义上说，我们可以说PCA的模式在测量尺度变化下是“阶稳定的”。一个特殊情况发生在沿某些模式正交的方向进行缩放时。此时，可能会出现明显的特征值交叉。然而，我们表明，我们可以将这些明显的交叉解释为模式瞬时交换其方向的情况，从而维持所需的阶稳定性。
英文摘要
The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's $k$-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's $k$-th largest principal component, for each $k$. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.
Comments9 pages, 6 figures
