Technical Report PHD-2019-12

Title: Learning for Numerical Geometry
Authors: Gautam Pai
Supervisors: Ron Kimmel
PDFCurrently accessibly only within the Technion network
Abstract: 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. Such end-to-end learning frameworks have been shown to generate state-of-the-art results in various areas of image processing and computer vision.

This thesis explores a synergy between these two disparate computational philosophies. In particular, we integrate deep learning into the computational methods of numerical geometry and propose neural network-based alternatives to standard geometric algorithms. We explore various ways of adopting deep-learning to work in conjunction, rather than replace, classical axiomatic approaches with an expectation that the net result is attributed with a degree of understanding and clarity in contrast to a black-box approach. With this motivation, the broad idea is to breakdown the computational scheme of a geometric algorithm and identify parts which can be intelligently replaced by neural networks and then propose a rediscovered numerical scheme with the learned constituents. Using this principle, we show that even for well-defined tasks that have established computational routines, a deep learned alternative yields improved results in terms of robustness and generalization to unseen examples.

First, we demonstrate that we can learn invariant geometric representations 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 exhibit improved numerical properties in comparison to their axiomatic counterparts.

Secondly, we demonstrate the method of 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 guided by a distance preserving loss. In addition to a straightforward out-of-sample extension, the MDS action of the network is shown to have superior generalization abilities.

Third, we develop a hybrid numerical scheme for computing geodesic distances on Euclidean grids as well as parametric surfaces by developing a Deep Eikonal Solver. We replace the formulaic local numerical solver in the well-known fast marching algorithm with a trained neural network, and we show that this leads to more accurate distance computations.

Finally, we tackle shape correspondence using descriptor dependent kernels in a functional maps framework. By aggregating the pairwise information from pointwise descriptors 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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