A Nash-type Dimensionality Reduction for Discrete Subsets of L2

Pursuing a line of work begun by Whitney, Nash showed that every C manifold of dimension d can be embedded in R in such a manner that the local structure at each point is preserved isometrically. We provide an analog of this result for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute near-isometric embedding of neighborhoods of bounded cardinality. There are a variety of empirical situations, abstracted to metric space tasks, in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widely-used heuristics such as Isomap and Locally Linear Embedding. In such situations we offer the possibility of dimension reductions unobtainable with global methods, because the dimension of our locally (1 + )-distorting embedding is proportional to −2 log k, where k is the cardinality of the neighborhoods where distances are preserved and is independent of the number of points in the metric space. One may view our work as a local version of the widely-used Johnson-Lindenstrauss lemma. We use a device of Nash, together with more recent metric embedding methods, to compose local dimensionality reducing embeddings within a global immersion (that preserves short distances), and with mild additional assumptions, a global embedding that also keeps distant points well-separated. We provide an efficient randomized algorithm for the embeddings. ∗[email protected]. School of Engineering and Computer Science, Hebrew University, Israel and Center for the Mathematics of Information, Caltech, CA, USA. Supported in part by a grant from the Israeli Science Foundation (195/02) and in part by a grant from the National Science Foundation (NSF CCF-065253). †[email protected]. Center for the Mathematics of Information, Caltech, Pasadena, CA 91125. ‡[email protected]. Caltech, Pasadena, CA 91125. Supported in part by NSA H98230-06-1-0074 and NSF CCF0515342.