Math @ Duke

Publications [#243810] of Mauro Maggioni
Papers Published
 Allard, WK; Chen, G; Maggioni, M, Multiscale geometric methods for data sets II: Geometric MultiResolution Analysis,
Applied and Computational Harmonic Analysis, vol. 32 no. 3
(May, 2012),
pp. 435462, Elsevier BV, ISSN 10635203 [doi]
(last updated on 2019/02/20)
Abstract: Data sets are often modeled as samples from a probability distribution in RD, for D large. It is often assumed that the data has some interesting lowdimensional structure, for example that of a ddimensional 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 RD (for example found by SVD), at a cost (n+D)d for n data points. When M is nonlinear, there are no "explicit" and algorithmically efficient constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by blackbox global optimization. In this paper we construct datadependent multiscale dictionaries that aim at efficiently encoding and manipulating the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa, in contrast with L1type sparsityseeking algorithms, but like adaptive nonlinear approximation in classical multiscale analysis. In addition, data points are guaranteed to have a compressible representation in terms of the dictionary, depending on the assumptions on the geometry of the underlying probability distribution. © 2011 Elsevier Inc. All rights reserved.


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