# Multi-Dimensional Scaling on Groups

## TL;DR

This paper investigates how symmetries in data influence the behavior of multi-dimensional scaling (MDS), revealing that understanding the symmetry group can predict MDS outcomes and identify fundamental components similar to Fourier analysis.

## Contribution

It introduces a theoretical framework linking data symmetries to MDS behavior, enabling pre-emptive insights into embeddings based on group properties.

## Key findings

- Symmetry groups determine MDS embedding structure.
- Only a few irreducible representations often suffice for accurate embedding.
- The approach parallels Fourier analysis in signal processing.

## Abstract

Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important in this area is understanding how symmetries inform the algorithms that we apply to our data. In this paper we explore the behavior of the dimensionality reduction algorithm multi-dimensional scaling (MDS) in the presence of symmetry. We show that understanding the properties of the underlying symmetry group allows us to make strong statements about the output of MDS even before applying the algorithm itself. In analogy to Fourier theory, we show that in some cases only a handful of fundamental "frequencies" (irreducible representations derived from the corresponding group) contribute information for the MDS Euclidean embedding.