Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis

TL;DR

This paper introduces fast, data-dependent multiscale dictionaries for efficient, sparse encoding of high-dimensional point cloud data, extending wavelet-like methods to nonlinear, low-dimensional structures.

Contribution

It develops novel multiscale dictionaries tailored for nonlinear manifolds, enabling efficient and sparse data representation with fast algorithms.

Findings

Constructed data-dependent multiscale dictionaries for point clouds.

Achieved fast encoding and decoding algorithms.

Guaranteed sparse representations for data points.

Abstract

Data sets are often modeled as point clouds in $R^D$, for $D$ large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a $d$-dimensional manifold $M$, with $d$ much smaller than $D$. When $M$ is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of $d$ vectors in $R^D$ (for example found by SVD), at a cost $(n+D)d$ for $n$ data points. When $M$ is nonlinear, there are no "explicit" constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box optimization. In this paper we construct data-dependent multi-scale dictionaries that aim at efficient encoding and manipulating of the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa. In addition, data points are guaranteed to have a sparse representation in terms of the dictionary. We think of dictionaries as the analogue of wavelets, but for approximating point clouds rather than functions.