We empirically investigate the effect of class manifold entanglement and the intrinsic and extrinsic dimensionality of the data distribution on the sample complexity of supervised classification with deep ReLU networks. We separate the effect of entanglement and intrinsic dimensionality and show statistically for artificial and real-world image datasets that the intrinsic dimensionality and the entanglement have an interdependent effect on the sample complexity. Low levels of entanglement lead to low increases of the sample complexity when the intrinsic dimensionality is increased, while for high levels of entanglement the impact of the intrinsic dimensionality increases as well. Further, we show that in general the sample complexity is primarily due to the entanglement and only secondarily due to the intrinsic dimensionality of the data distribution.

In this invited talk, we discuss state of the art in neural network verification. We propose the term continuous verification to characterise the family of methods that explore continuous nature of machine learning algorithms. We argue that methods of continuous verification must rely on robust programming language infrastructure (refinement types, automated proving, type-driven program synthesis), which provides a major opportunity for the declarative programming language community. Keywords: Neural Networks, Verification, AI.