Invariants of gradient (Morse) complexes and artificial neural nets

There is canonical partition of set of critical values of function into pairs "birth-death" and a separate set representing basis in Betti homology, as was established in S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994). This partition arises from bringing the gradient (Morse) complex to so called "canonical form" by a linear transform respecting the filtration defined by the order of the critical values. These "canonical forms" are combinatorial invariants of R-filtered complexes. Starting from the beginning of 2000s these invariants became widely used in applied mathematics under the name of "Persistence diagrams" and "Persistence Bar-codes". The canonical form of an R-filtered complex is the partition of the set of critical values (indices of filtration) into "birth-death" pairs and a separate set representing homology of the complex. The "canonical form" invariants are obtained by action of upper-triangular matrices, which reduces the R-filtered complex to the simplest form. In this note I give a short introduction into the subject of these invariants. I also propose application of these invariants to artificial neural nets