##### Department of Mathematics,

University of California San Diego

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### Data Science Seminar

## Facundo Memoli

## Classical Multidimensional scaling of metric measure spaces

##### Abstract:

We study a generalization of the classical Multidimensional Scaling procedure (cMDS) to the setting of general metric measure spaces which can be seen as natural 'continuous limits' of finite data sets. We identify certain crucial spectral properties of the generalized cMDS operator thus providing a natural and rigorous formulation of cMDS in this setting. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces such as high dimensional spheres and tori (both with their geodesic distance). In particular, the case of spheres (with geodesic distance) requires that we establish that its cMDS operator is *trace class*, a condition which is natural in the context when the cMDS operator has infinite rank. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov-Wasserstein distance.

### May 17, 2022

### 11:00 AM

SDSC E230

https://ucsd.zoom.us/j/93593146777

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