  Randomized smoothing has been successfully applied in high-dimensional image
classification tasks to obtain models that are provably robust against input
perturbations of bounded size. We extend this technique to produce certifiable
robustness for vector-valued functions, i.e., bound the change in output caused
by a small change in input. These functions are used in many areas of machine
learning, such as image reconstruction, dimensionality reduction,
super-resolution, etc., but due to the enormous dimensionality of the output
space in these problems, generating meaningful robustness guarantees is
difficult. We design a smoothing procedure that can leverage the local,
potentially low-dimensional, behaviour of the function around an input to
obtain probabilistic robustness certificates. We demonstrate the effectiveness
of our method on multiple learning tasks involving vector-valued functions with
a wide range of input and output dimensionalities.
