Classification on Manifolds Public Deposited

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Last Modified
  • March 20, 2019
Creator
  • Sen, Suman Kumar
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
Abstract
  • This dissertation studies classification on smooth manifolds and the behavior of High Dimensional Low Sample Size (HDLSS) data as the dimension increases. In modern image analysis, statistical shape analysis plays an important role in understanding several diseases. One of the ways to represent three dimensional shapes is the medial representation, the parameters of which lie on a smooth manifold, and not in the usual d-dimensional Euclidean space. Existing classification methods like Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD) do not naturally handle data lying on manifolds. We present a general framework of classification for data lying on manifolds and then extend SVM and DWD as special cases. The approach adopted here is to find control points on the manifold which represent the different classes of data and then define the classifier as a function of the distances (geodesic distances on the manifold) of individual points from the control points. Next, using a deterministic behavior of Euclidean HDLSS data, we show that the generalized version of SVM behaves asymptotically like the Mean Difference method as the dimension increases. Lastly, we consider the manifold (S2)d, and show that under some conditions, data lying on such a manifold has a deterministic geometric structure similar to Euclidean HDLSS data, as the dimension (number of components d in (S2)d) increases. Then we show that the generalized version of SVM behaves like the Geodesic Mean Difference (extension of the Mean Difference method to manifold data) under the deterministic geometric structure.
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  • In Copyright
Advisor
  • Marron, James Stephen
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  • Open access
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