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| Publications [#327596] of Mauro Maggioni
Papers Published
- Chen, G; Little, AV; Maggioni, M, Multi-resolution geometric analysis for data in high dimensions,
in Applied and Numerical Harmonic Analysis, vol. 1
(January, 2013),
pp. 259-285, Birkhäuser Boston, ISBN 9780817683757 [doi]
(last updated on 2019/04/10)
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 high-dimensional 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 low-dimensional smooth manifold. Samples are often corrupted by high-dimensional noise. We are interested in developing tools for studying the geometry of such high-dimensional data sets. In particular, we present here a multiscale transform that maps high-dimensional 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 multi-resolution analysis in wavelet theory: whereas wavelets map a signal (typically low dimensional, such as a one-dimensional time series or a two-dimensional image) to a set of multiscale coefficients, the geometric wavelets discussed here map points in a high-dimensional point cloud to a multiscale set of coefficients. The geometric multi-resolution 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 data-adaptive 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 real-world data sets.
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