We study the dynamics of an online algorithm for learning a sparse leading
eigenvector from samples generated from a spiked covariance model. This
algorithm combines the classical Oja's method for online PCA with an
element-wise nonlinearity at each iteration to promote sparsity. In the
high-dimensional limit, the joint empirical measure of the underlying sparse
eigenvector and its estimate provided by the algorithm is shown to converge
weakly to a deterministic, measure-valued process. This scaling limit is
characterized as the unique solution of a nonlinear PDE, and it provides exact
information regarding the asymptotic performance of the algorithm. For example,
performance metrics such as the cosine similarity and the misclassification
rate in sparse support recovery can be obtained by examining the limiting
dynamics. A steady-state analysis of the nonlinear PDE also reveals an
interesting phase transition phenomenon. Although our analysis is asymptotic in
nature, numerical simulations show that the theoretical predictions are
accurate for moderate signal dimensions.

研究基于在线 PCA 和逐元素非线性的算法对于稀疏主特征向量的学习，证明其在高维极限下收敛于确定性的，尺度测度过程，同时证明在稀疏支持恢复中有良好的表现，并通过非线性 PDE 的稳态分析揭示了有趣的相变现象。