Thouless-Anderson-Palmer Equations Research Articles

Thouless–Anderson–Palmer Equations for the Multi-species Sherrington–Kirkpatrick Model

Thouless–Anderson–Palmer Equations for the Multi-species Sherrington–Kirkpatrick Model

On the concavity of the TAP free energy in the SK model

We analyze the Hessian of the Thouless–Anderson–Palmer (TAP) free energy for the Sherrington–Kirkpatrick model, below the de Almeida–Thouless line, evaluated in Bolthausen’s approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, and that the support includes zero on the AT line. In this “macroscopic” sense, we show that TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetizations. This proves a spectral interpretation of the AT line. We also find different magnetizations than Bolthausen’s approximate solutions at which the Hessian of the TAP free energy has positive outlier eigenvalues. In particular, when the magnetizations are assumed to be independent of the disorder, we prove that Plefka’s second condition is equivalent to all eigenvalues being negative. On this occasion, we extend the convergence result of Capitaine et al. (Electron. J. Probab. 16, no. 64, 2011) for the largest eigenvalue of perturbed complex Wigner matrices to the GOE.

Thouless–Anderson–Palmer Equations for the Ghatak–Sherrington Mean Field Spin Glass Model

We derive the Thouless-Anderson-Palmer (TAP) equations for the Ghatak and Sherrington model. Our derivation, based on the cavity method, holds at high temperature and at all values of the crystal field. It confirms the prediction of Yokota.

Dynamical Approach to the TAP Equations for the Sherrington–Kirkpatrick Model

We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.

Thouless–Anderson–Palmer equations for generic $p$-spin glasses

We study the Thouless–Anderson–Palmer (TAP) equations for spin glasses on the hypercube. First, using a random, approximately ultrametric decomposition of the hypercube, we decompose the Gibbs measure, $\langle \cdot \rangle_{N}$, into a mixture of conditional laws, $\langle \cdot \rangle_{\alpha,N}$. We show that the TAP equations hold for the spin at any site with respect to $\langle \cdot \rangle_{\alpha,N}$ simultaneously for all $\alpha $. This result holds for generic models provided that the Parisi measure of the model has a jump at the top of its support.

Parameter inference in a probabilistic model using clustered data

We propose a method to infer the parameters of a probabilistic model from given data samples. Our method is based on the pseudolikelihood and composite likelihood methods. We cluster the given data samples and apply the clustered data samples to the pseudolikelihood and composite likelihood methods. From an expansion of the pseudolikelihood method around the mean of a cluster, the mean-field and Thouless–Anderson–Palmer equations are derived. Likewise, from an expansion of the composite likelihood method around the mean of a cluster, a method that is similar to the Bethe approximation is derived. We then perform numerical simulations using our method. We find that our method gives an accurate estimate in the range of weak coupling parameters but has an inferior accuracy compared to the pseudolikelihood and composite likelihood methods in the range of strong coupling parameters. In the range of strong coupling parameters, as the number of clusters increases, the inference accuracy of our method improves. Compared to the pseudolikelihood and composite likelihood methods, our method reduces the number of computational tasks for the estimation, therefore, sacrificing the inference accuracy.

Weighted community detection and data clustering using message passing

Grouping objects into clusters based on the similarities or weights between them is one of the most important problems in science and engineering. In this work, by extending message-passing algorithms and spectral algorithms proposed for an unweighted community detection problem, we develop a non-parametric method based on statistical physics, by mapping the problem to the Potts model at the critical temperature of spin-glass transition and applying belief propagation to solve the marginals corresponding to the Boltzmann distribution. Our algorithm is robust to over-fitting and gives a principled way to determine whether there are significant clusters in the data and how many clusters there are. We apply our method to different clustering tasks. In the community detection problem in weighted and directed networks, we show that our algorithm significantly outperforms existing algorithms. In the clustering problem, where the data were generated by mixture models in the sparse regime, we show that our method works all the way down to the theoretical limit of detectability and gives accuracy very close to that of the optimal Bayesian inference. In the semi-supervised clustering problem, our method only needs several labels to work perfectly in classic datasets. Finally, we further develop Thouless–Anderson–Palmer equations which heavily reduce the computation complexity in dense networks but give almost the same performance as belief propagation.

The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential

We present a parallel derivation of the Thouless–Anderson–Palmer (TAP) equations and of an effective thermodynamic potential for the negative perceptron and soft sphere models in high dimension. Both models are continuous constrained satisfaction problems with a critical jamming transition characterized by the same exponents. Our analysis reveals that a power expansion of the potential up to the second order constitutes a successful framework to approach the jamming points from the SAT phase (the region of the phase diagram where at least one configuration verifies all the constraints), where the ground-state energy is zero. An interesting outcome is that approaching the jamming line the effective thermodynamic potential has a logarithmic contribution, which turns out to be dominant in a proper scaling regime. Our approach is quite general and can be directly applied to other interesting models. Finally we study the spectrum of small harmonic fluctuations in the SAT phase recovering the typical scaling below the cutoff frequency but a different behavior characterized by a non-trivial exponent above it.

Quantum Thouless-Anderson-Palmer equations for glassy systems

We derive Thouless-Anderson-Palmer (TAP) equations for quantum disordered systems. We apply them to the study of the paramagnetic and glassy phases of a quantum extension of the spherical p spin-glass model. We generalize several useful quantities (complexity, threshold level, etc.) and various ideas (configurational entropy crisis, etc.) that have been developed within the classical TAP approach to quantum systems. The analysis of the quantum TAP equations allows us to show that the phase diagram (temperature versus quantum parameter) of the p spin-glass model should be generic. In particular, we argue that a crossover from a second-order thermodynamic transition close to the classical critical point to a first-order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems.

Thouless-anderson-palmer equations for neural networks

Previous derivation of the Thouless-Anderson-Palmer (TAP) equations for the Hopfield model by the cavity method yielded results that were inconsistent with those of the perturbation theory as well as the results derived by the replica theory of the model. Here we present a derivation of the TAP equation for the Hopfield model by the cavity method and show that it agrees with the form derived by perturbation theory. We also use the cavity method to derive TAP equations for the pseudoinverse neural network model. These equations are consistent with the results of the replica theory of these models.

Dynamical TAP approach to mean field glassy systems

The Thouless, Anderson, Palmer (TAP) approach to thermodynamics of mean field spin-glasses is generalised to dynamics. A method to compute the dynamical TAP equations is developed and applied to the p-spin spherical model. In this context we show to what extent the dynamics can be represented as an evolution in the free energy landscape. In particular the relationship between the long-time dynamics and the local properties of the free energy landscape shows up explicitly within this approach. Conversely, by an instantaneous normal modes analysis we show that the local properties of the energy landscape seen by the system during its dynamical evolution do not change qualitatively at the dynamical transition.

Fermionic Thouless-Anderson-Palmer equations

We derive the Thouless-Anderson-Palmer (TAP) equations for the fermionic Ising spin glass. It is found that, just as in the classical Sherrington-Kirkpatrick spin-glass model, the conditions for stability and for validity of the free energy are equivalent. We determine the breakdown of the paramagnetic phase. Numerical solutions of the fermionic TAP equations at T = 0 allowed us to localize a first-order transition between the spin-glass phase and the paramagnetic phase at µ≈0.8. We computed at zero temperature the filling factor ν(µ) and the distribution of the internal fields. The saddle-point equations resulting from the calculation of the number of solutions to the TAP equations were found to be much more complicated, as in the classical case.

Some developments in the theory of modulated order. I. The role of fluctuations in the axial next-nearest-neighbor Ising model and the relevance of the Thouless-Anderson-Palmer equation.

The general mean-field theory of systems with modulated order is extended so as to include nontrivial fluctuation effects through the use of the Kirkwood approximation. Application of the theory to the axial next-nearest-neighbor Ising (ANNNI) model reveals that fluctuations (1) decrease the size of the entire ordered region of the phase diagram, (2) decrease that fraction of the ordered region that is most highly modulated, and (3) suppress the reentrant parts of the modulated region that do survive. Physical reasons for this behavior are discussed. In addition, we note that our ANNNI mean-field equations turn out to be precisely the Thouless-Anderson-Palmer (TAP) equations of spin-glass theory despite a number of rather different assumptions made in deriving the TAP equations. Some of what this observation reveals about the TAP equations is discussed.