  Euclidean embedding from noisy observations containing outlier errors is an
important and challenging problem in statistics and machine learning. Many
existing methods would struggle with outliers due to a lack of detection
ability. In this paper, we propose a matrix optimization based embedding model
that can produce reliable embeddings and identify the outliers jointly. We show
that the estimators obtained by the proposed method satisfy a non-asymptotic
risk bound, implying that the model provides a high accuracy estimator with
high probability when the order of the sample size is roughly the degree of
freedom up to a logarithmic factor. Moreover, we show that under some mild
conditions, the proposed model also can identify the outliers without any prior
information with high probability. Finally, numerical experiments demonstrate
that the matrix optimization-based model can produce configurations of high
quality and successfully identify outliers even for large networks.
