UNIQUENESS OF WEIGHTS FOR RECURRENT NETS

with A ∈ IRn×n, B ∈ IRn×m, C ∈ IRp×n, and where ~σ(x) = (σ(x1), . . . , σ(xn)), with σ a function from IR to itself. (For the continuous-time case, we always assume that the function σ is globally Lipschitz, so that the existence and uniqueness of solutions for the differential equation is guaranteed.) We will consider the parameter identifiability problem, which asks about the possibility of recovering the entries of the matrices A, B, and C from the input/output map u(·) 7→ y(·) of the system. This question has been already addressed in [1] and in [2] (and, for feedforward nets, in [6] and in [4]) where it is proved that, under appropriate minimality assumptions, the zero-initial state i/o behavior determines, up to a small number of symmetries, the weights of the model. In this paper we establish the same result for arbitrary-initial state i/o maps. Moreover, we show that, for a generic subclass of these models, the minimality conditions needed in order for the results to hold are exactly the observability conditions found in the recent paper [3]. It is interesting to notice that, inside this subclass, these observability conditions are also necessary for identifiability.