Restricted isometric compression of sparse datasets into low-dimensional varieties

TL;DR

This paper extends the theory of restricted isometric projections from sparse datasets to low-dimensional varieties, enabling efficient dimensionality reduction for complex structured datasets.

Contribution

It introduces a new framework for restricted isometric compression of low-dimensional varieties, generalizing previous results from sparse datasets.

Findings

Achieves low-distortion embeddings for varieties of high codimension

Applicable to structured and hierarchical datasets

Extends isometric projection theory to complex data structures

Abstract

This article extends the known restricted isometric projection of sparse datasets in Euclidean spaces $\mathbb{R}^N$ down into low-dimensional subspaces $\mathbb{R}^k, k \ll N,$ to the case of low-dimensional varieties $\mathcal{M} \subset \mathbb{R}^N,$ of codimension $N - k = \omega(N)$. Applications to structured/hierarchical datasets are considered.