weakness
The main assumption behind the development of the theory is the fact that Lipchitz neural networks are "often perceived as not expressive", which I do not remember ever reading in the literature. Most of the paper seems fuzzy: it goes from BCE to robustness certification to a (trivial) convergence proposition (prop 4) to consideration about float32/64 (ex 1)... it is just too much. There are no transitions and at no moment the reader understands what is the point.
One of the biggest problem is the writing style that makes the paper hard and annoying (sorry...) to read. For example among many other problems:
- sentences must contain a verb and end with a '.'(even when it ends with a formula);
- theorem should be self contained, exterior references should remain exceptional;
- organize the equations in order to be easily readable;
- separate definitions and remarks (e.g. def 1 & 2 contains definitions and remarks);
- separate propositions and definitions (e.g. corollary 1 contains a definition and a proposition);
- the proofs rely on many technical tools such as Lebesgue measure but the pre-requisite (Borel space etc) are never explicitely given.
Most of the proofs in the appendix should be re-written. More personal but I think the footnotes in the paper could be avoided. The paper is full of references and it costs the readability of the arguments. The reader gets overwhelmed very quickly for the wrong reasons. References
most reference are incomplete (many article are reference as arxiv instead than the published and peer-reviewed version) Others
l94: one needs additional hypothesis on AllNet to claim that they have finite Lipschitz constant (Lipschitz activation would do).
l216: the reference to cor 1 seems dubious: it is a definition of \epsilon separation rather than about bias.
- separate definitions and remarks (e.g. def 1 & 2 contains definitions and remarks);