Sharp Asymptotics of Matrix Sketching for a Rank-One Spiked Model

We consider matrix sketching for principal component analysis (PCA) with the input data matrices generated by the rank-one spiked model. In the high-dimensional limit, we evaluate the estimation performance of matrix sketching via the replica method. Numerical studies confirm the validity of our results. The obtained result shows that the performance of the estimator undergoes a phase transition at a certain value of the signal strength. A similar asymptotic behavior is well-known for PCA. We demonstrate that our result is a one-parameter generalization of the existing results for PCA. On the basis of our performance evaluation, we also derive the condition for matrix sketching to recover the underlying signal.