Hehe. Yes that’s right, in the limit you can just analyse the singular values and vectors by hand, it’s nice.
No general implied connection to phase transitions, but the conjecture is that if there are phase transitions in your development then you can for general reasons expect PCA to “attempt” to use the implicit “coordinates” provided by the Lissajous curves (i.e. a binary tree, the first Lissajous curve uses PC2 to split the PC1 range into half, and so on) to locate stages within the overall development. I got some way towards proving that by extending the literature I cited in the parent, but had to move on, so take the story with a grain of salt. This seems to make sense empirically in some cases (e.g. our paper).
Hehe. Yes that’s right, in the limit you can just analyse the singular values and vectors by hand, it’s nice.
No general implied connection to phase transitions, but the conjecture is that if there are phase transitions in your development then you can for general reasons expect PCA to “attempt” to use the implicit “coordinates” provided by the Lissajous curves (i.e. a binary tree, the first Lissajous curve uses PC2 to split the PC1 range into half, and so on) to locate stages within the overall development. I got some way towards proving that by extending the literature I cited in the parent, but had to move on, so take the story with a grain of salt. This seems to make sense empirically in some cases (e.g. our paper).