Aim: This study aims to enhance interpretability and explainability of multi-modal prediction models integrating imaging and tabular patient data. Methods: We adapt the xAI methods Grad-CAM and Occlusion to multi-modal, partly interpretable deep transformation models (dTMs). DTMs combine statistical and deep learning approaches to simultaneously achieve state-of-the-art prediction performance and interpretable parameter estimates, such as odds ratios for tabular features. Based on brain imaging and tabular data from 407 stroke patients, we trained dTMs to predict functional outcome three months after stroke. We evaluated the models using different discriminatory metrics. The adapted xAI methods were used to generated explanation maps for identification of relevant image features and error analysis. Results: The dTMs achieve state-of-the-art prediction performance, with area under the curve (AUC) values close to 0.8. The most important tabular predictors of functional outcome are functional independence before stroke and NIHSS on admission, a neurological score indicating stroke severity. Explanation maps calculated from brain imaging dTMs for functional outcome highlighted critical brain regions such as the frontal lobe, which is known to be linked to age which in turn increases the risk for unfavorable outcomes. Similarity plots of the explanation maps revealed distinct patterns which give insight into stroke pathophysiology, support developing novel predictors of stroke outcome and enable to identify false predictions. Conclusion: By adapting methods for explanation maps to dTMs, we enhanced the explainability of multi-modal and partly interpretable prediction models. The resulting explanation maps facilitate error analysis and support hypothesis generation regarding the significance of specific image regions in outcome prediction.

During neural network training, the sharpness of the Hessian matrix of the training loss rises until training is on the edge of stability. As a result, even nonstochastic gradient descent does not accurately model the underlying dynamical system defined by the gradient flow of the training loss. We use an exponential Euler solver to train the network without entering the edge of stability, so that we accurately approximate the true gradient descent dynamics. We demonstrate experimentally that the increase in the sharpness of the Hessian matrix is caused by the layerwise Jacobian matrices of the network becoming aligned, so that a small change in the network preactivations near the inputs of the network can cause a large change in the outputs of the network. We further demonstrate that the degree of alignment scales with the size of the dataset by a power law with a coefficient of determination between 0.74 and 0.98.