Two Embedding Theorems for Data with Equivalences under Finite Group Action

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, nonsequential data spaces with equivalences under group action, for which analogues of two wellstudied embedding theorems are derived: the Whitney embedding theorem and the JohnsonLindenstrauss 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). Our first theorem shows that almost all such two-step embeddings can one-to-one embed the canonical part of a bounded, discriminable set of data points, if embedding dimension exceeds 2k whereby k is the box-counting dimension of the set closure of canonical data points. Our second theorem shows for k equal to the number of canonical points of a finite data set, a randomly sampled two-step embedding, preserves isometries (of the canonical part) up to factors 1 ± with probability at least 1− β, if the embedding dimension exceeds (2 log k + log(1/β))/α( , δ) for some function α, and δ is a positive constant capturing a certain discriminability property of the invariant. In the second theorem, the value k is tied only to the canonical part, which may be significantly smaller than the ambient data dimension, up to a factor equal to the size the group. ∗F. Lim recieved support from NSF Grant ECCS-1128226. ar X iv :1 20 7. 69 86 v2 [ cs .D S] 1 5 O ct 2 01 2