We study principal component analysis (PCA) under memory constraints, a setting that is increasingly important in large-scale data analysis. Our focus is on Oja's algorithm, which is a one-pass, memory-efficient algorithm requiring only $O(p)$ storage in the rank-one case and $O(pk)$ storage for $k$- PCA. The main goal is to develop a procedure based on Oja's algorithm that is optimal without prior knowledge of the eigengap, which is otherwise needed to tune the learning rates effectively. We do this by first introducing a geometric perspective on the convergence of $k$-PCA: by embedding the Oja iterates into the exterior space, we establish an exact equivalence between $k$-PCA in the ambient space and $1$-PCA in the exterior space. This geometric viewpoint enables us to establish several convergence guarantees, including one based only on a single learning-rate schedule. Building on this geometric perspective, we develop an averaging theory for $k$-PCA and show that the resulting averaged estimator is adaptive: it achieves the nearly optimal convergence rate without prior knowledge of the eigengap. As an application, we construct memory-efficient estimators for elliptical component analysis (ECA) \cite{han2014scale,han2018eca}. Simulation studies and real data analysis are conducted to demonstrate the benefits of our proposed algorithms.

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