  Multidimensional scaling is an important dimension reduction tool in
statistics and machine learning. Yet few theoretical results characterizing its
statistical performance exist, not to mention any in high dimensions. By
considering a unified framework that includes low, moderate and high
dimensions, we study multidimensional scaling in the setting of clustering
noisy data. Our results suggest that, the classical multidimensional scaling
can be modified to further improve the quality of embedded samples, especially
when the noise level increases. To this end, we propose {\it modified
multidimensional scaling} which applies a nonlinear transformation to the
sample eigenvalues. The nonlinear transformation depends on the dimensionality,
sample size and moment of noise. We show that modified multidimensional scaling
followed by various clustering algorithms can achieve exact recovery, i.e., all
the cluster labels can be recovered correctly with probability tending to one.
Numerical simulations and two real data applications lend strong support to our
proposed methodology.
