Asymptotic Analysis of Deep Residual Networks
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature: one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. The scaling regime one ends up with depends on certain features of the network architecture, such as the smoothness of the activation function. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit. In the case where the scaling limit is a stochastic differential equation, the deep network limit is shown to be described by a system of forward-backward stochastic differential equations. Joint work with: Alain-Sam Cohen (InstaDeep Ltd), Alain Rossier (Oxford), RenYuan Xu (University of Southern California).
Joint Reconstruction-Segmentation with Graph PDEs
In most practical image segmentation tasks, the image to be segmented will need to first be reconstructed from indirect, damaged, and/or noisy observations. Traditionally, this reconstruction-segmentation task would be done in sequence: first apply the reconstruction method, and then the segmentation method. Joint reconstruction-segmentation is a method for using segmentation and reconstruction techniques simultaneously, to use information from the segmentation to guide the reconstruction, and vice versa. Past work on this has employed relatively simple segmentation algorithms, such as the Chan–Vese algorithm. In this talk, we will demonstrate how joint reconstruction-segmentation can be done using the graph-PDE-based segmentation techniques developed by Bertozzi & Flenner (2012) and Merkurjev, Kostic, & Bertozzi (2013), with ideas drawn from Budd & van Gennip (2020) and Budd, van Gennip, & Latz (2021).