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Classical multidimensional scaling only works well when the noisy distancesobserved in a high dimensional space can be faithfully represented by Euclideandistances in a low dimensional space. Advanced models such as Maximum VarianceUnfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-DefiniteProgramming (SDP) to reconstruct such faithful representations. While those SDPmodels are capable of producing high quality configuration numerically, theysuffer two major drawbacks. One is that there exist no theoretically guaranteedbounds on the quality of the configuration. The other is that they are slow incomputation when the data points are beyond moderate size. In this paper, wepropose a convex optimization model of Euclidean distance matrices. Weestablish a non-asymptotic error bound for the random graph model withsub-Gaussian noise, and prove that our model produces a matrix estimator ofhigh accuracy when the order of the uniform sample size is roughly the degreeof freedom of a low-rank matrix up to a logarithmic factor. Our resultspartially explain why MVU and MVE often work well. Moreover, we develop a fastinexact accelerated proximal gradient method. Numerical experiments show thatthe model can produce configurations of high quality on large data points thatthe SDP approach would struggle to cope with.

Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction