Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey

Abstract

To appear as a part of an upcoming academic book on dimensionality reduction and manifold learning.

Clustering acoustic emission data streams with sequentially appearing clusters using mixture models

“…To conclude this experimental section while opening up some perspectives, we illustrate in Figure 20 (right-hand side) the output of non-metric multidimensional scaling (MDS), using Sammon's nonlinear mapping criterion [60,61], applied to the means found by GMMSEQ with 14 clusters on campaign B. MDS allows us to visualize, in a two-dimensional space, how close the means are to each other. It is computed from the pairwise Euclidean distances between the means.…”

Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Survey

“…Multidimensional Scaling (MDS) tries to preserve the distance after projection onto its subspace (Cox & Cox, 2008;Ghojogh et al, 2020b). We saw in Proposition 2 that metric learning can be seen as projection onto the column space of U where W = U U .…”

Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey

“…where D ∈ R n×n is a matrix with squared Euclidean distance between points and R n×n H := I − (1/n)11 is the centering matrix. Isomap also applies multidimensional scaling with a geodesic kernel which uses piece-wise Euclidean distance for computing D (Tenenbaum et al, 2000;Ghojogh et al, 2020b). The row summation of this kernel matrix is (Yan et al, 2005;2006):…”

“…( 6), the kernel used in kernel classical MDS or the geodesic kernel used in Isomap can be interpreted as the Laplacian matrix of graph of data as it satisfies its row-sum property. The optimization of MDS or Isomap is (Ghojogh et al, 2020b):…”