# 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…

## 8 Citations

### Infinite multidimensional scaling for metric measure spaces

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2022

For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space…

### Classical MDS on Metric Measure Spaces

- Mathematics
- 2022

. We study a generalization of the classical Multidimensional Scaling procedure (cMDS) to the setting of general metric measure spaces. We identify certain crucial spectral properties of the…

### Classical Multidimensional Scaling on Metric Measure Spaces

- Mathematics, Computer Science
- 2022

We study a generalization of the classical Multidimensional Scaling procedure (cMDS) which is applicable in the setting of general metric measure spaces which can be seen as natural “continuous…

### Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

- Mathematics, Computer Science
- 2020

An easy variational formulation of this problem is provided which leads to an algorithm always providing an almost isometric imbedding with given controlled small distortion of original distances.

### METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES

- MathematicsMathematika
- 2019

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere.…

### The Persistent Homology of Cyclic Graphs

- MathematicsInternational Journal of Computational Geometry & Applications
- 2022

We give an [Formula: see text] algorithm for computing the [Formula: see text]-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on [Formula: see text] vertices.…

### Research Statement: Bridging applied and quantitative topology

- Mathematics
- 2022

Henry Adams, Colorado State University Large sets of high-dimensional data are common in most branches of science, and their shapes reflect important patterns within. The goal of topological data…

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