This paper is concerned with the problem of representing and learning a linear transformation using a linear neural network. In recent years, there is a growing interest in the study of such networks, in part due to the successes of deep learning. The main question of this body of research (and also of our paper) is related to the existence and optimality properties of the critical points of the mean-squared loss function. An additional primary concern of our paper pertains to the robustness of these critical points in the face of (a small amount of) regularization. An optimal control model is introduced for this purpose and a learning algorithm (backprop with weight decay) derived for the same using the Hamilton's formulation of optimal control. The formulation is used to provide a complete characterization of the critical points in terms of the solutions of a nonlinear matrix-valued equation, referred to as the characteristic equation. Analytical and numerical tools from bifurcation theory are used to compute the critical points via the solutions of the characteristic equation.
Amirhossein Taghvaei (University of Illinois at Urbana-Champaign)
Amirhossein Taghvaei graduated from Sharif University of Technology, Tehran, Iran, in 2013, receiving two B.S degrees; in Mechanical Engineering and in Physics. He is currently a PhD student in the Mechanical Science and Engineering department at University of Illinois at Urbana-Champaign. He is working under the direction of Prof. Prashant Mehta in the Coordinated Science Laboratory, Decision and Control group. His research interest lies in the intersection of probability and pde. He is currently working on the Feedback Particle Filter algorithm, and its application to high dimensional problems.
Jin W Kim (University of Illinois)
Prashant Mehta (University of Illinois)
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