  A fundamental bottleneck in utilising complex machine learning systems for
critical applications has been not knowing why they do and what they do, thus
preventing the development of any crucial safety protocols. To date, no method
exist that can provide full insight into the granularity of the neural
network's decision process. In the past, saliency maps were an early attempt at
resolving this problem through sensitivity calculations, whereby dimensions of
a data point are selected based on how sensitive the output of the system is to
them. However, the success of saliency maps has been at best limited, mainly
due to the fact that they interpret the underlying learning system through a
linear approximation. We present a novel class of methods for generating
nonlinear saliency maps which fully account for the nonlinearity of the
underlying learning system. While agreeing with linear saliency maps on simple
problems where linear saliency maps are correct, they clearly identify more
specific drivers of classification on complex examples where nonlinearities are
more pronounced. This new class of methods significantly aids interpretability
of deep neural networks and related machine learning systems. Crucially, they
provide a starting point for their more broad use in serious applications,
where 'why' is equally important as 'what'.
