Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.4.2022.2.05
Volume 4, Issue 2, 1 August 2022, Pages 175-201

 

Abstract. Training a neural network for semantic segmentation with many images and pixel-level segmentations is a well-established computer-vision technique. When pixel-level segmentations are unavailable, weak supervision or prior information like bounding boxes and the size/shape of objects still enables training a network. Directly including prior knowledge on the segmentations means constraining the network output. This complicates the possible optimization strategies because the regularization then acts on the non-linear neural-network function output and not on the optimization variables. We present a new algorithm to include prior information via constraints on the network output, implemented via projection-based point-to-set distance functions, that are differentiable and always have the same functional form for the derivative. Thus, there is no need to adapt penalty functions or algorithms to various constraints. The distance function’s differentiability also avoids issues related to constraining properties typically associated with non-differentiable penalties. We show that by explicitly taking a general neural network structure into account, the Lagrangian for the problem `naturally’ decouples the constraints and neural network, which allows for a gradient computation via backpropagation/adjoint-state as is common in deep learning. We present a suite of constraint sets suitable for segmentation problems. The numerical experiments show that learning from constraint sets applies to the broader imaging sciences, including visual and non-visual imagery, even when training a network for a single example.

 

How to Cite this Article:
B. Peters, Point-to-set distance functions for output-constrained neural networks, J. Appl. Numer. Optim. 4 (2022), 175-201.