Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

14 Regularization in Neural Nets

Richard Szeliski Microsoft Corporation


1. INTRODUCTION

Regularization is a class of mathematical techniques used in data analysis and engineering to help solve difficult, ill-conditioned estimation and design problems. Regularization was originally applied to problems in numerical analysis, such as function approximation, and in statistics. The central idea in regularization is to restrict the range of possible solutions to ensure a unique and stable solution. Usually, this is achieved by imposing smoothness constraints on the solution, typically through a penalty term involving derivatives of the solution.

Regularization has been used successfully in computer vision for formulating and solving many low-level vision problems, such as stereo matching. It has strong connections to iterative and massively parallel (neural network) algorithms. Recently, regularization has been applied to the formulation and solution of learning problems in neural nets. In this chapter, we review the applications of regularization in computer vision and neural networks and demonstrate how regularization provides a coherent framework and a useful set of techniques for solving problems that arise in these fields.

The chapter is organized around three main themes: regularization applied to computer vision, the Bayesian interpretation of regularization, and regularization applied to learning in neural networks. We begin the chapter with a review of the classical formulation of regularization applied to the solution of inverse problems, that is, problems where only the sampled output of a model or process are given and its unknown parameters must be recovered. We use one-dimensional curve

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