# The Rare Eclipse Problem on Tiles: Quantised Embeddings of Disjoint Convex Sets

## TL;DR

This paper investigates conditions under which quantised random embeddings preserve the separability of disjoint convex sets, enabling exact classification after dimensionality reduction, with theoretical results and numerical validation.

## Contribution

It provides a new theoretical framework relating embedding dimension, quantiser resolution, and set separation for preserving separability in quantised embeddings.

## Key findings

- Derived conditions linking embedding parameters and set separation.
- Numerical phase transition curves for two -balls.
- Experimental validation of theoretical results.

## Abstract

Quantised random embeddings are an efficient dimensionality reduction technique which preserves the distances of low-complexity signals up to some controllable additive and multiplicative distortions. In this work, we instead focus on verifying when this technique preserves the separability of two disjoint closed convex sets, i.e., in a quantised view of the "rare eclipse problem" introduced by Bandeira et al. in 2014. This separability would ensure exact classification of signals in such sets from the signatures output by this non-linear dimensionality reduction. We here present a result relating the embedding's dimension, its quantiser resolution and the sets' separation, as well as some numerically testable conditions to illustrate it. Experimental evidence is then provided in the special case of two $\ell_2$-balls, tracing the phase transition curves that ensure these sets' separability in the embedded domain.