Task-Driven Wavelets using Constrained Empirical Risk Minimization

Deep Neural Networks (DNNs) are widely used for their ability to effectively approximate large classes of functions. This flexibility, however, makes the strict enforcement of constraints on DNNs a difficult problem. In contexts where it is critical to limit the function space to which certain network components belong, such as wavelets employed in Multi-Resolution Analysis (MRA), naive constraints via additional terms in the loss function are inadequate. To address this, we introduce a Convolutional Neural Network (CNN) wherein the convolutional filters are strictly constrained to be wavelets. This allows the filters to update to task-optimized wavelets during the training procedure. Our primary contribution lies in the rigorous formulation of these filters via a constrained empirical risk minimization framework, thereby providing an exact mechanism to enforce these structural constraints. While our work is grounded in theory, we investigate our approach empirically through applications in medical imaging, particularly in the task of contour prediction around various organs, achieving superior performance compared to baseline methods.