Physics Based Supervised and Unsupervised Learning of Graph Structure

Ayan T Sinha, Purdue University

Abstract

Graphs are central tools to aid our understanding of biological, physical, and social systems. Graphs also play a key role in representing and understanding the visual world around us, 3D-shapes and 2D-images alike. In this dissertation, I propose the use of physical or natural phenomenon to understand graph structure. I investigate four phenomenon or laws in nature: (1) Brownian motion, (2) Gauss's law, (3) feedback loops, and (3) neural synapses, to discover patterns in graphs. Random walks is the mathematical formalization of Brownian motion on graphs. I discuss the connection between the Laplace-Beltrami operator governing the diffusion equation and the transition matrix of a random walk. This connection is used to construct multiscale shape signatures with several desirable properties, and applied to shape matching, segmentation and search. Next, I establish a connection between the Laplacian of a general graph and the flux of a potential field over a boundary. This insight is exactly an extension of Gauss’ law for charge distributions applied to discrete graphs. I design a novel method for data clustering using this law in conjunction with laws of conservation and spectral graph theory. My method to identify community boundaries is significantly more robust to noise than traditional detection of strongly knit modules. Feedback is pervasive in natural as well as artificial systems. I design a new method to separate feedback effects in recommender systems such as Netflix from intrinsic preferences of users. This method is based on a simple deconvolution of an iterative feedback loop common in physical systems that admit feedback. I am able to identify items recommended to a user by the recommender system by simply inputting the user-item ratings matrix to my algorithm. Finally, I study artificial neural networks inspired by neural synapses in the human brain. I develop a technique to create a geometric image from a point cloud, so that convolutional neural networks used pervasively for image understanding can be used for 3D shape understanding. I construct deep neural models which use the parametric knowledge of point clouds from a large database to parameterize unknown and incomplete point clouds. This bridges the gap between traditional geometric processing and data-driven supervised image processing techniques. Furthermore, I demonstrate how pooled activations of neurons in the networks can be used for real-time understanding of point cloud data. I infer the hand skeleton structure from depth images as a prototypical application. Overall, I show how interesting structures in networks, shapes and images can be inferred using physical laws of nature.

Degree

Ph.D.

Advisors

Ramani, Purdue University.

Subject Area

Applied Mathematics|Mechanical engineering|Artificial intelligence

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