Multidimensional scaling on metric measure spaces

@article{Adams2020MultidimensionalSO,
  title={Multidimensional scaling on metric measure spaces},
  author={Henry Adams and Mark Blumstein and Lara Kassab},
  journal={Rocky Mountain Journal of Mathematics},
  year={2020},
  volume={50},
  pages={397-413}
}
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into $\mathbb{R}^m$ for all $m… 

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References

SHOWING 1-10 OF 30 REFERENCES

Multidimensional Scaling: Infinite Metric Measure Spaces

A notion of MDS on infinite metric measure spaces is studied, along with its optimality properties and goodness of fit, and questions on convergence of M DS are addressed.

HORSESHOES IN MULTIDIMENSIONAL SCALING AND LOCAL KERNEL METHODS

Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

  • F. Mémoli
  • Computer Science
    Found. Comput. Math.
  • 2011
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers.

Data Visualization With Multidimensional Scaling

This article discusses methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis, and shows applications to the mapping of computer usage data, to the dimension reduction of marketing segmentation data,to the layout of mathematical graphs and social networks, and finally to the spatial reconstruction of molecules.

Learning Eigenfunctions Links Spectral Embedding and Kernel PCA

In this letter, we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem: learning the

Multidimensional scaling.

Key aspects of performing MDS are discussed, such as methods that can be used to collect similarity estimates, analytic techniques for treating proximity data, and various concerns regarding interpretation of the MDS output.

13 Theory of multidimensional scaling

  • J. LeeuwW. Heiser
  • Computer Science
    Classification, Pattern Recognition and Reduction of Dimensionality
  • 1982

Convex Optimization & Euclidean Distance Geometry

This book is about convex optimization, convex geometry (with particular attention to distance geometry), geometric problems, and problems that can be transformed into geometrical problems.

Random matrix approximation of spectra of integral operators

~H n, obtained by deleting its diagonal. It is proved that the l 2 distance between the ordered spectrum of Hn and the ordered spectrum of H tends to zero a.s. if and only if H is Hilbert‐Schmidt.

The Past, Present, and Future of Multidimensional Scaling

This paper pays tribute to several important developers of M DS and gives a subjective overview of milestones in MDS developments and discusses the present situation of MDS and give a brief outlook on its future.