Spectral multidimensional scaling

Abstract

Significance Scientific theories describe observations by equations with a small number of parameters or dimensions. Memory and computational efficiency of dimension reduction procedures is a function of the size of the observed data. Sparse local operators that involve almost linear complexity, and faithful multiscale models with quadratic cost, make the design of dimension reduction tools a delicate balance between modeling accuracy and efficiency. Here, we propose multiscale modeling of the data at a modest computational cost. We project the classical multidimensional scaling problem into the data spectral domain. Then, embedding into a low-dimensional space is efficiently accomplished. Theoretical support and empirical evidence demonstrate that working in the natural eigenspace of the data, one could reduce the complexity while maintaining model fidelity.