  In this paper, we study the problem of principal component analysis with
generative modeling assumptions, adopting a general model for the observed
matrix that encompasses notable special cases, including spiked matrix recovery
and phase retrieval. The key assumption is that the underlying signal lies near
the range of an $L$-Lipschitz continuous generative model with bounded
$k$-dimensional inputs. We propose a quadratic estimator, and show that it
enjoys a statistical rate of order $\sqrt{\frac{k\log L}{m}}$, where $m$ is the
number of samples. We also provide a near-matching algorithm-independent lower
bound. Moreover, we provide a variant of the classic power method, which
projects the calculated data onto the range of the generative model during each
iteration. We show that under suitable conditions, this method converges
exponentially fast to a point achieving the above-mentioned statistical rate.
We perform experiments on various image datasets for spiked matrix and phase
retrieval models, and illustrate performance gains of our method to the classic
power method and the truncated power method devised for sparse principal
component analysis.
