Spectral LPM: An Optimal Locality-Preserving Mapping using the Spectral (not Fractal) Order

For the past two decades, fractals (e.g., the Hilbert and Peano space-filling curves) have been considered the natural method for providing a locality-preserving mapping. The idea behind a locality-preserving mapping is to map points that are nearby in the multi-dimensional space into points that are nearby in the one-dimensional space. In this paper, we argue against the use of fractals in locality-preserving mapping algorithms, and present examples with experimental evidence to show why fractals produce poor locality-preserving mappings. In addition, we propose an optimal locality-preserving mapping algorithm, termed the Spectral Locality-Preserving Mapping algorithm (Spectral LPM, for short), that makes use of the spectrum of the multi-dimensional space. We give a mathematical proof for the optimality of Spectral LPM, and also demonstrate its practical use.