Prof. Erik Aurell (KTH)
The "Inverse Ising Problem" refers to finding the parameters (the J_ij's and the h_i's) in an Ising model given the first and second moments (the magnitizations m_i and the correclation functions c_ij). This is of great interest in machine learning and data analysis whenever the data set and the number of variables is large, but the values taken by the variables can be taken to be "high" and "low". The maximum entropy distributions with given first and second moments then has the Ising form where the h_i's and J_ij's are Lagrange parameters. The last years have seen an explosion in interest in approximate but fast methods borrowed from statistical mechanics to learn such "maxentropy" models from correlation data. Some motivations have been e.g. inferring casual structures underlying observed gene expression, or inferring functional connectivities between neurons from multi-neuronal recordings, where measurements from hundreds of neurons are available today, and millions have been envisaged. Although methods borrowed from non-equilibrium may be more promising in applications, I will describe results using equilibrium statistical mechanics, and the testing ground will be mainly the Sherrington-Kirkpatrick spin glass. The methods discussed are simple mean-field, TAP, and the "Susceptibility Propagation" introduced by Mezard. One main message is that all these are sensitive to the accuracy of the correlation data themselves. There is hence a three-way trade-off between computability, inference accuracy (given perfect data), and sensitivity to undersampling of the correlations. This is work done or in progess with John Hertz, Yasser Roudi, Mikko Alava, Hamed Mahmoudi, Aymeric Fouquier d'Herouel, Jarkko Salojärvi, Zeng Hong-Li and Charles Ollion. Similar results to ours on Susceptibility Propagation have been obtained by Enzo Marinari (paper available on arXiv.org).