Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning: most deep learning methods require the solution of hard optimisation problems, and a good understanding of the tradeoff between computational effort, amount of data and model complexity is required to successfully design a deep learning approach for a given problem. A large amount of progress made in deep learning has been based on heuristic explorations, but there is a growing effort to mathematically understand the structure in existing deep learning methods and to systematically design new deep learning methods to preserve certain types of structure in deep learning. In this article, we review a number of these directions: some deep neural networks can be understood as discretisations of dynamical systems, neural networks can be designed to have desirable properties such as invertibility or group equivariance, and new algorithmic frameworks based on conformal Hamiltonian systems and Riemannian manifolds to solve the optimisation problems have been proposed. We conclude our review of each of these topics by discussing some open problems that we consider to be interesting directions for future research.
U-Nets have been established as a standard neural network design architecture for image-to-image learning problems such as segmentation and inverse problems in imaging. For high-dimensional applications, as they for example appear in 3D medical imaging, U-Nets however have prohibitive memory requirements. Here, we present a new fully-invertible U-Net-based architecture called the \emph{iUNet}, which allows for the application of highly memory-efficient backpropagation procedures. For this, we introduce learnable and invertible up- and downsampling operations. An open source library in Pytorch for 1D, 2D and 3D data is made available.
Neural networks have recently been established as a viable classification method for imaging mass spectrometry data for tumor typing. For multi-laboratory scenarios however, certain confounding factors may strongly impede their performance. In this work, we introduce Deep Relevance Regularization, a method of restricting what the neural network can focus on during classification, in order to improve the classification performance. We demonstrate how Deep Relevance Regularization robustifies neural networks against confounding factors on a challenging inter-lab dataset consisting of breast and ovarian carcinoma. We further show that this makes the relevance map -- a way of visualizing the discriminative parts of the mass spectrum -- sparser, thereby making the classifier easier to interpret
In recent years, an increasing number of neural network models have included derivatives with respect to inputs in their loss functions, resulting in so-called double backpropagation for first-order optimization. However, so far no general description of the involved derivatives exists. Here, we cover a wide array of special cases in a very general Hilbert space framework, which allows us to provide optimized backpropagation rules for many real-world scenarios. This includes the reduction of calculations for Frobenius-norm-penalties on Jacobians by roughly a third for locally linear activation functions. Furthermore, we provide a description of the discontinuous loss surface of ReLU networks both in the inputs and the parameters and demonstrate why the discontinuities do not pose a big problem in reality.
Recent studies on the adversarial vulnerability of neural networks have shown that models trained to be more robust to adversarial attacks exhibit more interpretable saliency maps than their non-robust counterparts. We aim to quantify this behavior by considering the alignment between input image and saliency map. We hypothesize that as the distance to the decision boundary grows,so does the alignment. This connection is strictly true in the case of linear models. We confirm these theoretical findings with experiments based on models trained with a local Lipschitz regularization and identify where the non-linear nature of neural networks weakens the relation.
Motivation: Tumor classification using Imaging Mass Spectrometry (IMS) data has a high potential for future applications in pathology. Due to the complexity and size of the data, automated feature extraction and classification steps are required to fully process the data. Deep learning offers an approach to learn feature extraction and classification combined in a single model. Commonly these steps are handled separately in IMS data analysis, hence deep learning offers an alternative strategy worthwhile to explore. Results: Methodologically, we propose an adapted architecture based on deep convolutional networks to handle the characteristics of mass spectrometry data, as well as a strategy to interpret the learned model in the spectral domain based on a sensitivity analysis. The proposed methods are evaluated on two challenging tumor classification tasks and compared to a baseline approach. Competitiveness of the proposed methods are shown on both tasks by studying the performance via cross-validation. Moreover, the learned models are analyzed by the proposed sensitivity analysis revealing biologically plausible effects as well as confounding factors of the considered task. Thus, this study may serve as a starting point for further development of deep learning approaches in IMS classification tasks.