有限群作用下等价数据的两个嵌入定理
Two Embedding Theorems for Data with Equivalences under Finite Group Action
摘要：
最近有兴趣在压缩数据集的非顺序设置，其中缺乏明显的排序对他们的数据空间，需要考虑的数据等价的概念。例如，Varshney & Goyal（DCC，2006）考虑了多集等价，而Choi & Szpankowski（IEEE Trans. IT，2012）考虑了图中的同构等价。在这里等价被认为是在一个相对广泛的框架下-有限维的，非序列的数据空间与等价下的组行动，其中类似的两个深入研究嵌入定理推导：惠特尼嵌入定理和约翰逊-林登施特劳斯引理。只有规范数据点需要仔细嵌入，每个这样的点代表一组数据点在群体作用下是等价的。两步嵌入被认为是。首先，应用群不变量来解释等价性，然后，线性嵌入将其降低到低维。我们的研究结果需要对所应用的不变量的可辨别性进行假设，例如与分离不变量相关的概念（Dufresne，2008）和模式识别中的完整性（Kakarala，1992）。在后一个定理中，嵌入复杂度取决于规范部分的大小，它可能比整个数据集小得多，直到一个因子等于组的大小。
英文摘要：
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.
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