On the typical properties of inverse problems in statistical mechanics
arXiv:1311.0190
Abstract
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).
PhD disseration presented to the Faculty of SISSA (Trieste, Italy) on September 17th 2012
References in corpus (5)
- High-dimensional Ising model selection using -regularized logistic regression
- The Ising Model for Neural Data: Model Quality and Approximate Methods for Extracting Functional Connectivity
- Small-correlation expansions for the inverse Ising problem
- Ising models for networks of real neurons
- Inference algorithms for gene networks: a statistical mechanics analysis