arXiv subjects
Publications and source records attributed to Victor Issa.
We identify the limit free energy of all multi-species spin glasses with centered $\pm 1$ spins. The result was previously known only under a convexity assumption on the covariance function of the Hamiltonian. We also obtain a one-species reduction of the formula for balanced multi-species models.
This paper constitutes the first part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.
This paper constitutes the second part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part does not satisfy the usual convexity assumption. In this case, the Parisi formula breaks down, and there are no known methods to fully identify the limit free energy. It was suggested in [arXiv:1906.08471] that the limit free energy may be described using the unique solution of a partial differential equation of Hamilton--Jacobi type. In the present paper, we prove the validity of this conjecture in the high-temperature regime and provide an explicit representation for the free energy in terms of critical points. Using the duality between the free energy and large deviation principles, one can then easily deduce from the previous result a large deviation principle for the mean magnetization as well as a representation for the free energy of spin glass models with additional Mattis interaction at high temperature. In the companion paper, we establish similar results at all temperatures for models whose spin glass part is assumed to satisfy the usual convexity assumption.
We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space.
In the PDE approach to mean-field spin glasses, it has been observed that the free energy of convex spin glass models could be enriched by adding an extra parameter in its definition, and that the thermodynamic limit of the enriched free energy satisfies a partial differential equation. This parameter can be thought of as a matrix-valued path, and the usual free energy is recovered by setting this parameter to be the constant path taking only the value $0$. Furthermore, the enriched free energy can be expressed using a variational formula, which is a natural extension of the Parisi formula for the usual free energy. For models with scalar spins the Parisi formula can be expressed as an optimization problem over a convex set, and it was shown in [arXiv:1402.5132] that this problem has a unique optimizer thanks to a strict convexity property. For models with vector spins, the Parisi formula cannot easily be written as a convex optimization problem. In this paper, we generalize the uniqueness of Parisi measures proven in [arXiv:1402.5132] to the enriched free energy of models with vector spins when the extra parameter is a strictly increasing path. Our approach relies on a Gateaux differentiability property of the free energy and the envelope theorem.
In this short note, we consider models of optimal Bayesian inference of finite-rank tensor products. We add to the model a linear channel parametrized by $h$. We show that at every interior differentiable point $h$ of the free energy (associated with the model), the overlap concentrates at the gradient of the free energy and the minimum mean-square error converges to a related limit. In other words, the model is replica-symmetric at every differentiable point. At any signal-to-noise ratio, such points $h$ form a full-measure set (hence $h=0$ belongs to the closure of these points). For a sufficiently low signal-to-noise ratio, we show that every interior point is a differentiable point.
We study mean-field spin glass models with general vector spins and convex covariance function. For those models, it is known that the limit of the free energy can be written as the supremum of a functional, this is the celebrated Parisi formula. In this paper, we observe that the Parisi functional extends into a concave and Lipschitz functional on the set of signed measures. We use this fact and Fenchel-Moreau duality to derive an un-inverted version of the Parisi formula. Namely, we show that the limit of the free energy can be written as the infimum of a functional related to the Parisi functional. This un-inverted formula can be interpreted as a Hopf-like formula for some Hamilton-Jacobi equation in Wasserstein space.
We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on the initial condition or the nonlinearity, and can therefore be utilized in contexts where the viscosity solution admits no standard variational representation.
It has recently been shown in [arXiv:2310.06745] that, upon constraining the system to stay in a balanced state, the Parisi formula for the mean-field Potts model can be written as an optimization problem over permutation-invariant functional order parameters. In this paper, we focus on permutation-invariant mean-field spin glass models. After introducing a correction term in the definition of the free energy and without constraining the system, we show that the limit free energy can be written as an optimization problem over permutation-invariant functional order parameters. We also show that for some models this optimization problem admits a unique optimizer. In the case of Ising spins, the correction term can be easily removed, and those results transfer to the uncorrected limit free energy. We also derive an upper bound for the limit free energy of some nonconvex permutation-invariant models. This upper bound is expressed as a variational formula and is related to the solution of some Hamilton-Jacobi equation. We show that if no first order phase transition occurs, then this upper bound is equal to the lower bound derived in [arXiv:2010.09114]. We expect that this hypothesis holds at least in the high temperature regime. Our method relies on the fact that the free energy of any convex mean-field spin glass model can be interpreted as the strong solution of some Hamilton-Jacobi equation.
It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.