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Multi-Resolution Geometric Analysis for Data in High Dimensions
| Content Provider | Semantic Scholar |
|---|---|
| Author | Chen, Guangliang Little, Anna V. Maggioni, Mauro |
| Copyright Year | 2013 |
| Abstract | 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. |
| Starting Page | 259 |
| Ending Page | 285 |
| Page Count | 27 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/978-0-8176-8376-4_13 |
| Alternate Webpage(s) | https://www.springer.com/cda/content/document/cda_downloaddocument/9780817683757-c1.pdf?SGWID=0-0-45-1366803-p174611839 |
| Alternate Webpage(s) | http://users.ju.edu/alittle2/Publications/Publication_ExcursionsInHarmonicAnalysisBookChapter.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/978-0-8176-8376-4_13 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
