H. Chau Nguyen, Johannes Berg
Taking a set of spin configurations sampled from the equilibrium distribution of an Ising model, can the underlying couplings between spins be reconstructed from a large number of such samples? This inverse Ising problem is a paradigmatic inverse problem with applications in neural biology, protein structure determination and gene expression analysis. Typically a large number of spins (representing neurons, genetic loci) is involved, as well as a large number of interactions between them. Mean-field approximations are often used to invert the relationship between the model parameters (external fields and couplings between spins) and observables (magnetisations and correlations), allowing to determine model parameters from data. However, all known mean-field methods fail at low temperatures. In this note, we show how clustering spin configurations can approximate thermodynamic states, and how mean-field methods applied to these thermodynamic states allow to reconstruct Ising models also at low temperatures.
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http://arxiv.org/abs/1204.5375
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