Theses and Dissertations from UMD

Permanent URI for this communityhttp://hdl.handle.net/1903/2

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a give thesis/dissertation in DRUM

More information is available at Theses and Dissertations at University of Maryland Libraries.

Browse

Filters

2010 - 201912020 - 20221

Settings

Subject: search.filters.subject.Dimension Reduction

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Topological Data Analysis, Dimension Reduction, and Computational Efficiency
    (2022) Monson, Nathaniel; Czaja, Wojciech; Brosnan, Patrick; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In this dissertation, we present a novel stability result for the persistent homology of the Rips complex associated to a point cloud. Our theorem is narrower than the classic result of Cohen-Steiner, Edelsbrunner, and Harer in that it does not apply to Cech complexes, nor to functions which are not measuring distance to a point cloud. It is broader than the classic result in that it is “local”; if a function approximately preserves distances in some range, but is contractionary below or expansionary above that range, our result still applies. The novel stability result is paired with the Johnson-Lindenstrauss Lemma to show that, with high probability, random projection approximately preserves persistent homology. An experimental analysis is given of the computational speedup granted by this dimension reduction. This is followed by some observations suggesting that even when the theoretical bound is loose enough that we have no guarantee of homology preservation, thereis still a high chance that significant features of the dataset are preserved.
  • Thumbnail Image
    Item
    The Multiplicative Zak Transform, Dimension Reduction, and Wavelet Analysis of LIDAR Data
    (2010) Flake, Justin Christopher; Benedetto, John J; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis broadly introduces several techniques within the context of timescale analysis. The representation, compression and reconstruction of DEM and LIDAR data types is studied with directional wavelet methods and the wedgelet decomposition. The optimality of the contourlet transform, and then the wedgelet transform is evaluated with a valuable new structural similarity index. Dimension reduction for material classification is conducted with a frame-based kernel pipeline and a spectral-spatial method using wavelet packets. It is shown that these techniques can improve on baseline material classification methods while significantly reducing the amount of data. Finally, the multiplicative Zak transform is modified to allow the study and partial characterization of wavelet frames.