Hans-Peter Beise

Hans-Peter Beise

We leverage the framework of hyperplane arrangements to analyze potential regions of (stable) fixed points. We provide an upper bound on the number of fixed points for multi-layer neural networks equipped with piecewise linear (PWL) activation functions with arbitrary many linear pieces. The theoretical optimality of the exponential growth in the number of layers of the latter bound is shown. Specifically, we also derive a sharper upper bound on the number of stable fixed points for one-hidden-layer networks with hard tanh activation.

Hans-Peter Beise, Steve Dias Da Cruz

In Radhakrishnan et al. [2020], the authors empirically show that autoencoders trained with usual SGD methods shape out basins of attraction around their training data. We consider network functions of width not exceeding the input dimension and prove that in this situation basins of attraction are bounded and their complement cannot have bounded components. Our conditions in these results are met in several experiments of the latter work and we thus address a question posed therein. We also show that under some more restrictive conditions the basins of attraction are path-connected. The tightness of the conditions in our results is demonstrated by means of several examples. Finally, the arguments used to prove the above results allow us to derive a root cause why scalar-valued neural network functions that fulfill our bounded width condition are not dense in spaces of continuous functions.

Jan Sokolowski, Volker Schulz, Udo Schröder et al.

In several studies, hybrid neural networks have proven to be more robust against noisy input data compared to plain data driven neural networks. We consider the task of estimating parameters of a mechanical vehicle model based on acceleration profiles. We introduce a convolutional neural network architecture that is capable to predict the parameters for a family of vehicle models that differ in the unknown parameters. We introduce a convolutional neural network architecture that given sequential data predicts the parameters of the underlying data's dynamics. This network is trained with two objective functions. The first one constitutes a more naive approach that assumes that the true parameters are known. The second objective incorporates the knowledge of the underlying dynamics and is therefore considered as hybrid approach. We show that in terms of robustness, the latter outperforms the first objective on noisy input data.

Steve Dias Da Cruz, Oliver Wasenmüller, Hans-Peter Beise et al.

We release SVIRO, a synthetic dataset for sceneries in the passenger compartment of ten different vehicles, in order to analyze machine learning-based approaches for their generalization capacities and reliability when trained on a limited number of variations (e.g. identical backgrounds and textures, few instances per class). This is in contrast to the intrinsically high variability of common benchmark datasets, which focus on improving the state-of-the-art of general tasks. Our dataset contains bounding boxes for object detection, instance segmentation masks, keypoints for pose estimation and depth images for each synthetic scenery as well as images for each individual seat for classification. The advantage of our use-case is twofold: The proximity to a realistic application to benchmark new approaches under novel circumstances while reducing the complexity to a more tractable environment, such that applications and theoretical questions can be tested on a more challenging dataset as toy problems. The data and evaluation server are available under https://sviro.kl.dfki.de.

Hans-Peter Beise, Steve Dias Da Cruz, Udo Schröder

We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approximation capabilities by [Hanin 2017 Approximating] and connectivity of decision regions by [Nguyen 2018 Neural] for such narrow neural networks. Our results are illustrated by means of numerical experiments.