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There is recent interest in compressing data sets for non-sequentialsettings, where lack of obvious orderings on their data space, require notionsof 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 areconsidered under a relatively broad framework - finite-dimensional,non-sequential data spaces with equivalences under group action, for whichanalogues of two well-studied embedding theorems are derived: the Whitneyembedding theorem and the Johnson-Lindenstrauss lemma. Only the canonical datapoints need to be carefully embedded, each such point representing a set ofdata points equivalent under group action. Two-step embeddings are considered.First, a group invariant is applied to account for equivalences, and thensecondly, a linear embedding takes it down to low-dimensions. Our resultsrequire hypotheses on discriminability of the applied invariant, such notionsrelated to seperating invariants (Dufresne, 2008), and completeness in patternrecognition (Kakarala, 1992). In the latter theorem, the embedding complexitydepends on the size of the canonical part, which may be significantly smallerthan the whole data set, up to a factor equal to the size the group.

Two Embedding Theorems for Data with Equivalences under Finite Group Action