Inverse Ising Problem Research Articles

The inverse Ising problem for one-dimensional chains with arbitrary finite-rangecouplings

We study Ising chains with arbitrary multispin finite-range couplings, providing an explicitsolution of the associated inverse Ising problem, i.e. the problem of inferring the values ofthe coupling constants from the correlation functions. As an application, we reconstruct thecouplings of chain Ising Hamiltonians having exponential or power-law two-spin plusthree- or four-spin couplings. The generalization of the method to ladders and to Isingsystems where a mean-field interaction is added to general finite-range couplings is alsodiscussed.

Message passing algorithms for the Hopfield network reconstruction: Threshold behavior and limitation

The Hopfield network is reconstructed as an inverse Ising problem by passing messages. The applied susceptibility propagation algorithm is shown to improve significantly on other mean-field-type methods and extends well into the low-temperature region. However, this iterative algorithm is limited by the nature of the supplied data. Its performance deteriorates as the data become highly magnetized, and this method finally fails in the presence of the frozen type data where at least two of its magnetizations are equal to 1 in absolute value. On the other hand, a threshold behavior is observed for the susceptibility propagation algorithm and the transition from good reconstruction to poor one becomes sharper as the network size increases.

Dynamics and performance of susceptibility propagation on synthetic data

We study the performance and convergence properties of the susceptibility propagation (SusP) algorithm for solving the Inverse Ising problem. We first study how the temperature parameter (T) in a Sherrington-Kirkpatrick model generating the data influences the performance and convergence of the algorithm. We find that at the high temperature regime (T > 4), the algorithm performs well and its quality is only limited by the quality of the supplied data. In the low temperature regime (T < 4), we find that the algorithm typically does not converge, yielding diverging values for the couplings. However, we show that by stopping the algorithm at the right time before divergence becomes serious, good reconstruction can be achieved down to T ≈ 2. We then show that dense connectivity, loopiness of the connectivity, and high absolute magnetization all have deteriorating effects on the performance of the algorithm. When absolute magnetization is high, we show that other methods can be work better than SusP. Finally, we show that for neural data with high absolute magnetization, SusP performs less well than TAP inversion.

Granger causality and the inverse Ising problem

The inference of the couplings of an Ising model with given means and correlations is called the inverse Ising problem. This approach has received a lot of attention as a tool to analyze neural data. We show that autoregressive methods may be used to learn the couplings of an Ising model, also in the case of asymmetric connections and for multispin interactions. We find that, for each link, the linear Granger causality is two times the corresponding transfer entropy (i.e., the information flow on that link) in the weak coupling limit. For sparse connections and a low number of samples, the ℓ 1 regularized least squares method is used to detect the interacting pairs of spins. Nonlinear Granger causality is related to multispin interactions.

Reconstructing the Hopfield network as an inverse Ising problem

We test four fast mean-field-type algorithms on Hopfield networks as an inverse Ising problem. The equilibrium behavior of Hopfield networks is simulated through Glauber dynamics. In the low-temperature regime, the simulated annealing technique is adopted. Although performances of these network reconstruction algorithms on the simulated network of spiking neurons are extensively studied recently, the analysis of Hopfield networks is lacking so far. For the Hopfield network, we found that, in the retrieval phase favored when the network wants to memory one of stored patterns, all the reconstruction algorithms fail to extract interactions within a desired accuracy, and the same failure occurs in the spin-glass phase where spurious minima show up, while in the paramagnetic phase, albeit unfavored during the retrieval dynamics, the algorithms work well to reconstruct the network itself. This implies that, as an inverse problem, the paramagnetic phase is conversely useful for reconstructing the network while the retrieval phase loses all the information about interactions in the network except for the case where only one pattern is stored. The performances of algorithms are studied with respect to the system size, memory load, and temperature; sample-to-sample fluctuations are also considered.

Small-correlation expansions for the inverse Ising problem

We present a systematic small-correlation expansion to solve the inverse Ising problem and find a set of couplings and fields corresponding to a given set of correlations and magnetizations. Couplings are calculated up to the third order in the correlations for generic magnetizations and to the seventh order in the case of zero magnetizations; in addition, we show how to sum some useful classes of diagrams exactly. The resulting expansion outperforms existing algorithms on the Sherrington–Kirkpatrick spin-glass model.

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity