  Randomized smoothing is currently considered the state-of-the-art method to
obtain certifiably robust classifiers. Despite its remarkable performance, the
method is associated with various serious problems such as "certified accuracy
waterfalls", certification vs.\ accuracy trade-off, or even fairness issues.
Input-dependent smoothing approaches have been proposed with intention of
overcoming these flaws. However, we demonstrate that these methods lack formal
guarantees and so the resulting certificates are not justified. We show that in
general, the input-dependent smoothing suffers from the curse of
dimensionality, forcing the variance function to have low semi-elasticity. On
the other hand, we provide a theoretical and practical framework that enables
the usage of input-dependent smoothing even in the presence of the curse of
dimensionality, under strict restrictions. We present one concrete design of
the smoothing variance function and test it on CIFAR10 and MNIST. Our design
mitigates some of the problems of classical smoothing and is formally
underlined, yet further improvement of the design is still necessary.
