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.

从包含异常值错误的嘈杂观测值中进行欧氏嵌入是统计和机器学习中一个重要且具有挑战性的问题。由于缺乏检测能力，许多现有方法将与异常值作斗争。在本文中，我们提出了一种基于矩阵优化的嵌入模型，该模型可以产生可靠的嵌入并共同识别异常值。我们表明，通过所提出的方法获得的估计量满足非渐近风险界限，这意味着当样本量的阶数大致为自由度直至对数因子时，该模型提供了具有高概率的高精度估计量。此外，我们表明，在某些温和条件下，所提出的模型还可以在没有任何先验信息的情况下以高概率识别异常值。最后，