Math @ Duke

Publications [#327596] of Mauro Maggioni
Papers Published
 Chen, G; Little, AV; Maggioni, M, Multiresolution geometric analysis for data in high dimensions,
in Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center, vol. 1
(January, 2013),
pp. 259285, Birkhaüser Boston, ISBN 9780817683764 [doi]
(last updated on 2018/02/21)
Abstract: © Springer Science+Business Media New York 2013. Large data sets arise in a wide variety of applications and are often modeled as samples from a probability distribution in highdimensional space. It is sometimes assumed that the support of such probability distribution is well approximated by a set of low intrinsic dimension, perhaps even a lowdimensional smooth manifold. Samples are often corrupted by highdimensional noise. We are interested in developing tools for studying the geometry of such highdimensional data sets. In particular, we present here a multiscale transform that maps highdimensional data as above to a set of multiscale coefficients that are compressible/sparse under suitable assumptions on the data. We think of this as a geometric counterpart to multiresolution analysis in wavelet theory: whereas wavelets map a signal (typically low dimensional, such as a onedimensional time series or a twodimensional image) to a set of multiscale coefficients, the geometric wavelets discussed here map points in a highdimensional point cloud to a multiscale set of coefficients. The geometric multiresolution analysis (GMRA) we construct depends on the support of the probability distribution, and in this sense it fits with the paradigm of dictionary learning or dataadaptive representations, albeit the type of representation we construct is in fact mildly nonlinear, as opposed to standard linear representations. Finally, we apply the transform to a set of synthetic and realworld data sets.


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