Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data

Abstract

We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution in R^D, that is nearly supported on a d-dimensional set M - for example supported on a d-dimensional Riemannian manifold. Geometric Multi-Resolution Analysis (GMRA) provides a robust and computationally efficient procedure to construct low-dimensional geometric approximations of M at varying resolutions. We introduce a thresholding algorithm on the geometric wavelet coefficients, leading to what we call adaptive GMRA approximations. We show that these data-driven, empirical approximations perform well, when the threshold is chosen as a suitable universal function of the number of samples n, on a wide variety of measures , that are allowed to exhibit different regularity at different scales and locations, thereby efficiently encoding data from more complex measures than those supported on manifolds. These approximations yield a data-driven dictionary, together with a fast transform mapping data to coefficients, and an inverse of such a map. The algorithms for both the dictionary construction and the transforms have complexity C n n with the constant linear in D and exponential in d. Our work therefore establishes adaptive GMRA as a fast dictionary learning algorithm with approximation guarantees. We include several numerical experiments on both synthetic and real data, confirming our theoretical results and demonstrating the effectiveness of adaptive GMRA.

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Reproductions