The phase-transition conjecture for weighted PCA recovery

Let $n$ and $d$ tend to infinity so that the sample-to-dimension ratio satisfies $n/d\to c>0$, and let the noise variance proportions satisfy $n_\ell/n\to p_\ell$ for $\ell=1,\dots,L$. For component $i$, let $A(\beta_i)$ denote the quantity governing the asymptotic recoveries, and let $\hat{u}_i,u_j,\hat{z}_i,z_j$, and the weighted inner product $\langle\cdot,\cdot\rangle_{\mathbf{W}^2}$ be as in the paper.

Phase-transition conjecture. If $A(\beta_i)\leq 0$, then

$$\sum_{j=1}^k|\langle\hat{u}_i,u_j\rangle|^2\overset{\mathrm{a.s.}}{\longrightarrow}0, \qquad \sum_{j=1}^k\left|\left\langle\frac{\hat{z}_i}{\sqrt n},\frac{z_j}{\sqrt n}\right\rangle_{\mathbf{W}^2}\right|^2\overset{\mathrm{a.s.}}{\longrightarrow}0.$$

Equivalently, the formulas for the asymptotic component and score recoveries extend to $A(\beta_i)\leq0$ by truncating $r_i^{(u)}$ and $r_i^{(z)}$ at zero.

This conjectures a phase transition at $A(\beta_i)=0$: beyond the threshold, both component and weighted score recoveries have zero asymptotic overlap. The cited asymptotic recovery formulas are stated only when $A(\beta_i)>0$, so the extension to the nonpositive regime remains open.