Speaker
Description
Comparing empirical subspaces of mismatched dimensions is a tricky problem that naturally arises in modern high-dimensional PCA, particularly within cross-validation and distributed learning frameworks. Traditional heuristics like the scree plot often yield ambiguous "elbows," leaving the true number of underlying signals up to guesswork.
In this talk, I will present the asymptotic geometric behavior between independent empirical subspaces. By tracking the chordal (or Wasserstein) distance across varying candidate dimensions, we introduce an alternative approach to PCA rank selection. Instead of a vague scree plot elbow, our geometric risk curve forms a distinct "Nike Swoosh". The definitive global minimum - the exact bottom of the swoosh—pinpoints the true number of significant signals.
To justify this visual heuristic, we formulate the exact deterministic macroscopic phase transition between independent empirical subspaces under data splitting. In a high-dimensional regime, we prove that a half-split is strictly optimal, yielding an explicit closed-form boundary for signal recovery. Practically, our "Nike Swoosh" risk curve provides a robust metric for consistent signal extraction, and the geometric distortion of the swoosh serves as a direct indicator of whether a given finite sample (usually a few hundred) has already fallen into the high-dimensional regime.