Two Embedding Theorems for Data with Equivalences under Finite Group Action

TL;DR

This paper develops two embedding theorems for data spaces with group-based equivalences, enabling dimension reduction while respecting data symmetries, with applications in non-sequential data compression.

Contribution

It introduces analogues of Whitney and Johnson-Lindenstrauss theorems for data under finite group actions, incorporating invariants and two-step embeddings.

Findings

Embedding complexity depends on the size of the canonical data subset.

Invariant-based embeddings can effectively handle data equivalences.

Theorems extend classical embedding results to symmetry-aware data spaces.

Abstract

There is recent interest in compressing data sets for non-sequential settings, where lack of obvious orderings on their data space, require notions of data equivalences to be considered. For example, Varshney & Goyal (DCC, 2006) considered multiset equivalences, while Choi & Szpankowski (IEEE Trans. IT, 2012) considered isomorphic equivalences in graphs. Here equivalences are considered under a relatively broad framework - finite-dimensional, non-sequential data spaces with equivalences under group action, for which analogues of two well-studied embedding theorems are derived: the Whitney embedding theorem and the Johnson-Lindenstrauss lemma. Only the canonical data points need to be carefully embedded, each such point representing a set of data points equivalent under group action. Two-step embeddings are considered. First, a group invariant is applied to account for equivalences, and then secondly, a linear embedding takes it down to low-dimensions. Our results require hypotheses on discriminability of the applied invariant, such notions related to seperating invariants (Dufresne, 2008), and completeness in pattern recognition (Kakarala, 1992). In the latter theorem, the embedding complexity depends on the size of the canonical part, which may be significantly smaller than the whole data set, up to a factor equal to the size the group.