Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches.

该研究提出了一种基于表面的谱广义多维尺度映射 (Spectral-GMDS) 方法，用于计算两个度量空间之间的映射关系，该方法在解决问题时隐含地将映射的平滑性纳入，从而实现了高效的嵌入，并在实验中取得了很好的效果。