High-dimensional data sets arising in a wide variety of applications often exhibit inherently low-dimensional structure. Detecting, measuring, and exploiting such low intrinsic dimensionality has been the focus of much research in the past decade, with implications and applications in many fields including high-dimensional statistics, machine learning, and signal processing. In this vein, active and compelling research in machine learning explores the topic of manifold learning, where the low-dimensional sets manifest as an unknown manifold structure that must be learned from the sampled data. Manifold learning seems quite distinct from the comparably popular subject of dictionary learning, where the low-dimensional structure is the set of sparse (or compressible) linear combinations of vectors from a finite linear dictionary. However, Geometric Multi-Resolution Analysis (Allard, Chen, and Maggioni, 2012) was introduced as a method for producing, in a robust multiscale fashion, an approximation to a low-dimensional manifold structure (should it exist), while simultaneously providing a dictionary for sparse representation of the data, thereby creating a connection between these two problems. In this work, we prove non-asymptotic probabilistic bounds for GMRA approximation error under certain assumptions on the geometry of the underlying distribution. In particular, our results imply that if the data is supported near a low-dimensional manifold, the proposed sparse representations result in an error primarily dependent upon the intrinsic dimension of the manifold, and independent of the ambient dimension.
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