A Stochastic Approach to Classification Error Estimates in Convolutional
  Neural Networks
Jan Peleska Felix Brüning Mario Gleirscher Wen-ling Huang
This technical report presents research results achieved in the field of
verification of trained Convolutional Neural Network (CNN) used for image
classification in safety-critical applications. As running example, we use the
obstacle detection function needed in future autonomous freight trains with
Grade of Automation (GoA) 4. It is shown that systems like GoA 4 freight trains
are indeed certifiable today with new standards like ANSI/UL 4600 and ISO 21448
used in addition to the long-existing standards EN 50128 and EN 50129.
Moreover, we present a quantitative analysis of the system-level hazard rate to
be expected from an obstacle detection function. It is shown that using
sensor/perceptor fusion, the fused detection system can meet the tolerable
hazard rate deemed to be acceptable for the safety integrity level to be
applied (SIL-3). A mathematical analysis of CNN models is performed which
results in the identification of classification clusters and equivalence
classes partitioning the image input space of the CNN. These clusters and
classes are used to introduce a novel statistical testing method for
determining the residual error probability of a trained CNN and an associated
upper confidence limit. We argue that this greybox approach to CNN
verification, taking into account the CNN model's internal structure, is
essential for justifying that the statistical tests have covered the trained
CNN with its neurons and inter-layer mappings in a comprehensive way.