Remco Duits (Eindhoven University of Technology)

Equivariant Deep Learning via PDEs on the Space of Positions and Orientations

We present PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) that generalize Group equivariant Convolutional Neural Networks (G-CNNs). In PDE-G-CNNs a network layer is a set of PDE-solvers where geometrically meaningful PDE-coefficients become trainable weights.

The underlying PDEs are morphological and linear scale space PDEs on the homogeneous space M of positions and orientations. They provide a geometrical and probabilistic PDE-design of the equivariant network. Equivariance means that roto-translation of input yields the same roto-translation of the output.

The network is implemented by morphological convolutions with approximations to kernels solving nonlinear HJB-PDEs (for morphological α-scale spaces), and to linear convolutions solving linear PDEs (for linear α-scale space).  In the morphological setting, the parameter α regulates soft max-pooling over balls, whereas in the linear setting the cases α=1/2 and α=1 correspond to the Poisson and Gaussian semigroup respectively.

We prove that our analytic approximation kernels are accurate and show that they are practical.  We build on a key isomorphism between the linear and nonlinear PDEs via the (approximate) Fourier-Cramér transform. This connects the underlying α-stable Lévy processes to Bellman processes on M. The equivariant PDE-G-CNN network implementation consists solely of linear and morphological convolutions on M. Common ad-hoc nonlinearities in CNNs are now obsolete and excluded.

We present blood vessel segmentation experiments in medical images that show clear benefits of PDE-G-CNNs compared to state-of-the-art G-CNNs: increase of performance along with a huge reduction in network parameters. We also present experiments that reveal clear benefits of the equivariant CNNs over normal CNNs on mitosis/cancer detection in medical images.

This is joint work of the presenter with B.M.N. Smets, J.W. Portegies, M. Veta (TU/e) and E.J. Bekkers (UvA).

Dirk van der Hoeven (Leiden University)

Exploiting the Surrogate Gap in Online Multiclass Classification

I will talk about the online multiclass classification setting with a surrogate loss. The setting proceeds in rounds 1 to T, and in each round the learner has to issue a prediction based on a d-dimensional feature vector, after which the learner either gets to see the correct answer (full information) or only whether he was right or wrong (bandit information). The goal of this sequential setting is to provide upper bounds on the number of mistakes the learner makes with respect to a surrogate loss, for example the hinge loss or the logistic loss. I will talk about the standard approach to control the number of mistakes and I will explain how one can improve upon the standard approach by exploiting the gap between the zero-one loss and the surrogate loss. In the full information version of the online multiclass classification setting one can improve the standard sqrt(T) bound to a bound that is independent of the number of rounds T. In the bandit version the standard bound depends on the dimension of the feature vector, which can be improved to a bound that is independent of the dimension of the feature vector.

Jelmer Wolterink (University of Twente)

Machine Learning on Graphs and Meshes for 3D Blood Vessel Analysis

Deep learning has had a tremendous impact on (medical) image analysis. Convolutional neural networks (CNNs) allow routine processing of medical images to extract e.g. high-quality triangular mesh segmentations of organs, or graphs representing the structure of vessel trees. A natural next step is to apply machine learning to these representations to extract additional clinically valuable information. However, current CNNs do not generalize to data that is not organized on a regular grid. In this talk, I will review how recent developments in graph and mesh convolutional networks can be leveraged to extract more structural, anatomical, and functional information from medical images, with applications in the improved analysis of 3D blood vessels.