How to interpret weights of the Principal Component Analysis of the Ising model?

Question

I'm trying to replicate the results obtained in this paper: https://arxiv.org/pdf/1606.00318.pdf . On page 3 the autors mention that the fact that the weight of the first principal component is uniform on all lattice sites means that "the transformation actually gives the uniform magnetization:"

$$m=\frac{1}{N}\sum_i\sigma_i$$

I have absolutely no idea of what this is supposed to mean. Is it supposed to be that the weight (intended as the eigenvalue of the principal eigenvector) is proportional to the magnetization? But how can we deduce that from the fact that the vector has uniform components?

Answer 1

The feature corresponding to the principal component for the $a$-th datapoint is $$Y_{a,1}=\sum_{j=1}^N X_{a,j}W_{j,1}$$ where $Y_{\bullet,l}\equiv y_l$ and $W_{\bullet,l}\equiv w_l$ in the notation of equation 4.

Since it is shown (empirically, in the inset of fig. 1) that all components of $w_1$ are approximately equal to some constant, call it $c$, then $$Y_{a,1}\approx c\sum_{j=1}^N X_{a,j}$$ which means that the most relevant feature for a configuration $a$ is something proportional to the magnetization $\sum_{i=1}^N\sigma_i^{(a)}$.

I'm not completely sure why $c=\frac1N$, but it should only be a matter of convention since $w_1$ clearly remains an eigenvector under re-scaling.