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Data sets are often modeled as point clouds in $R^D$, for $D$ large. It isoften assumed that the data has some interesting low-dimensional structure, forexample 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 forencoding 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 thatachieve a similar efficiency: typically one uses either random dictionaries, ordictionaries obtained by black-box optimization. In this paper we constructdata-dependent multi-scale dictionaries that aim at efficient encoding andmanipulating of the data. Their construction is fast, and so are the algorithmsthat map data points to dictionary coefficients and vice versa. In addition,data points are guaranteed to have a sparse representation in terms of thedictionary. We think of dictionaries as the analogue of wavelets, but forapproximating point clouds rather than functions.

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