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Learning for Numerical Geometry
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Gautam Pai, Ph.D. Thesis Seminar
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Wednesday, 14.8.2019, 11:00
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Taub 401 Taub Bld.
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Advisor:  Prof. Ron Kimmel
Numerical geometry comprises of principled computational methods that utilize theoretical insights from geometry along with the engineering concepts from numerical methods, for tackling various problems in geometric data analysis. In contrast, computational methods from recent advances in deep learning exhibit a black box nature where essential and meaningful features are learned from examples of training data leading to state-of-the-art results. This thesis explores a synergy between these two disparate computational philosophies. In particular, we integrate deep learning into computational methods of numerical geometry and propose neural network based alternatives to standard geometric algorithms. First, we demonstrate that we can learn an invariant geometric representation of planar curves using deep metric learning with a binary contrastive loss. Using just positive and negative examples of transformations, we show that a convolutional neural network is able to model an invariant function of a discrete planar curve and that such invariants show improved numerical properties in comparison to their axiomatic counterparts. Secondly, we demonstrate a scheme for deep isometric manifold learning for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of high dimensional manifolds. We use the philosophy of Multidimensional Scaling to train a network using the distance preserving loss in a manifold learning setup. In addition to a straightforward out-of-sample extension, MDS action of the network is shown to have superior generalization abilities. Finally, we tackle shape correspondence using descriptor dependent kernels in a functional maps framework. We interpret such kernels as operators of functions defined on compact two dimensional Riemannian manifolds. By aggregating the pairwise information from the descriptor and the intrinsic geometry of the surface encoded in the heat kernel, we construct a hybrid kernel and call it the bilateral operator. By forcing the correspondence map to commute with the Bilateral operator, we show that we can maximally exploit the information from a given set of point-wise descriptors in a functional map framework.
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