  Dimension reduction and data visualization aim to project a high-dimensional
dataset to a low-dimensional space while capturing the intrinsic structures in
the data. It is an indispensable part of modern data science, and many
dimensional reduction and visualization algorithms have been developed.
However, different algorithms have their own strengths and weaknesses, making
it critically important to evaluate their relative performance for a given
dataset, and to leverage and combine their individual strengths. In this paper,
we propose an efficient spectral method for assessing and combining multiple
visualizations of a given dataset produced by diverse algorithms. The proposed
method provides a quantitative measure -- the visualization eigenscore -- of
the relative performance of the visualizations for preserving the structure
around each data point. Then it leverages the eigenscores to obtain a consensus
visualization, which has much improved { quality over the individual
visualizations in capturing the underlying true data structure.} Our approach
is flexible and works as a wrapper around any visualizations. We analyze
multiple simulated and real-world datasets from diverse applications to
demonstrate the effectiveness of the eigenscores for evaluating visualizations
and the superiority of the proposed consensus visualization. Furthermore, we
establish rigorous theoretical justification of our method based on a general
statistical framework, yielding fundamental principles behind the empirical
success of consensus visualization along with practical guidance.
