  Optimization within a layer of a deep-net has emerged as a new direction for
deep-net layer design. However, there are two main challenges when applying
these layers to computer vision tasks: (a) which optimization problem within a
layer is useful?; (b) how to ensure that computation within a layer remains
efficient? To study question (a), in this work, we propose total variation (TV)
minimization as a layer for computer vision. Motivated by the success of total
variation in image processing, we hypothesize that TV as a layer provides
useful inductive bias for deep-nets too. We study this hypothesis on five
computer vision tasks: image classification, weakly supervised object
localization, edge-preserving smoothing, edge detection, and image denoising,
improving over existing baselines. To achieve these results we had to address
question (b): we developed a GPU-based projected-Newton method which is
$37\times$ faster than existing solutions.
