Multidimensional scaling of noisy high dimensional data

Multidimensional Scaling (MDS) is a classical technique for embedding data in low dimensions, still widespread use today. In this paper we study MDS modern setting - specifically, high dimensions and ambient measurement noise. We show that as the noise level increases, suffers sharp breakdown depends on dimension level, derive an explicit formula point case of white then introduce MDS+, simple variant MDS, which applies shrinkage nonlinearity to eigenvalues similarity matrix. Under natural loss function measuring quality, prove MDS+ unique, asymptotically optimal function. offers improved embedding, sometimes significantly so, compared with MDS. Importantly, calculates dimension, into should be embedded.