  In this work we consider the problem of extracting a set of interaction
parameters from an high-dimensional dataset describing T independent
configurations of a complex system composed of N binary units. This problem is
formulated in the language of statistical mechanics as the problem of finding a
family of couplings compatible with a corresponding set of empirical
observables in the limit of large N. We focus on the typical properties of its
solutions and highlight the possible spurious features which are associated
with this regime (model condensation, degenerate representations of data,
criticality of the inferred model). We present a class of models (complete
models) for which the analytical solution of this inverse problem can be
obtained, allowing us to characterize in this context the notion of stability
and locality. We clarify the geometric interpretation of some of those aspects
by using results of differential geometry, which provides means to quantify
consistency, stability and criticality in the inverse problem. In order to
provide simple illustrative examples of these concepts we finally apply these
ideas to datasets describing two stochastic processes (simulated realizations
of a Hawkes point-process and a set of time-series describing financial
transactions in a real market).
