Bulk Johnson-Lindenstrauss Lemmas
Michael P. Casey

TL;DR
This paper extends the Johnson-Lindenstrauss lemma by showing that if only a fraction of distances are allowed to be distorted, a lower target dimension suffices, especially for data with certain stable rank properties.
Contribution
It introduces a bulk JL lemma that allows a fraction of distances to be distorted, reducing the target dimension based on stable rank and subset properties.
Findings
A target dimension of O(ε^{-2} log(4e/η) log(N)/R) suffices when only a fraction η of distances are distorted.
Stable rank of matrices influences the dimension reduction bounds.
Refined results are provided for i.i.d. random vectors, especially when isotropic.
Abstract
For a set of points in , the Johnson-Lindenstrauss lemma provides random linear maps that approximately preserve all pairwise distances in -- up to multiplicative error with high probability -- using a target dimension of . Certain known point sets actually require a target dimension this large -- any smaller dimension forces at least one distance to be stretched or compressed too much. What happens to the remaining distances? If we only allow a fraction of the distances to be distorted beyond tolerance , we show a target dimension of is sufficient for the remaining distances. With the stable rank of a matrix as , the parameter is the minimal stable rank over certain sized subsets of or their…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Computational Geometry and Mesh Generation · Stochastic Gradient Optimization Techniques
