A Nonlinear Approach to Dimension Reduction

TL;DR

This paper proves that snowflake metrics of doubling sets in Euclidean space can be embedded into low-dimensional Euclidean space with near-isometric distortion, advancing dimension reduction techniques based on intrinsic data properties.

Contribution

It establishes new dimension reduction bounds for snowflake metrics of doubling sets, with embeddings into low-dimensional spaces depending only on the doubling constant.

Findings

Snowflake metrics of doubling sets embed into low-dimensional l2 with near-constant distortion.

Target dimension is polylogarithmic in the doubling constant.

Techniques extend to l1 and l_infinity spaces, with quantitative differences.

Abstract

The $l_2$ flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03,MatousekProblems07,ABN08,CGT10]), and is still open. We prove another result in this line of work: The snowflake metric $d^{1/2}$ of a doubling set $S \subset l_2$ embeds with constant distortion into $l_2^D$, for dimension $D$ that depends solely on the doubling constant of the metric. In fact, the distortion can be made arbitrarily close to 1, and the target dimension is polylogarithmic in the doubling constant. Our techniques are robust and extend to the more difficult spaces $l_1$ and $l_\infty$, although the dimension bounds here are quantitatively inferior than those for $l_2$.