  We propose a one-step procedure to estimate the latent positions in random
dot product graphs efficiently. Unlike the classical spectral-based methods
such as the adjacency and Laplacian spectral embedding, the proposed one-step
procedure takes advantage of both the low-rank structure of the expected
adjacency matrix and the Bernoulli likelihood information of the sampling model
simultaneously. We show that for each vertex, the corresponding row of the
one-step estimator converges to a multivariate normal distribution after proper
scaling and centering up to an orthogonal transformation, with an efficient
covariance matrix. The initial estimator for the one-step procedure needs to
satisfy the so-called approximate linearization property. The one-step
estimator improves the commonly-adopted spectral embedding methods in the
following sense: Globally for all vertices, it yields an asymptotic sum of
squares error no greater than those of the spectral methods, and locally for
each vertex, the asymptotic covariance matrix of the corresponding row of the
one-step estimator dominates those of the spectral embeddings in spectra. The
usefulness of the proposed one-step procedure is demonstrated via numerical
examples and the analysis of a real-world Wikipedia graph dataset.
