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Eliran Subag

Publications and source records attributed to Eliran Subag.

At least 19 recordsLinked to original sources
arXiv2026

Geometry of spherical spin glasses

Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and their relationship to the Gibbs measure. For the pure models, the measure concentrates on spherical bands around critical points that approximately maximize the energy at a particular radius. Next, we present another approach in which a similar picture is derived for general mixed models. At the core of this approach is a free energy functional computed over bands using multiple orthogonal replicas, satisfying a strong concentration of measure. We discuss several implications of this method for a generalized Thouless-Anderson-Palmer (TAP) approach. Finally, we explain how these geometric insights inform optimization algorithms, and briefly relate them to Smale's 17th problem over the real numbers.

math.PR↗
arXiv2025

Dynamics for spherical spin glasses: Gibbs distributed initial conditions

We derive the coupled non-linear integro-differential equations for the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics at temperature $T$, for spherical mixed $p$-spin disordered mean-field models, initialized according to a Gibbs measure for temperature $T_0$, in the replica-symmetric (RS) or $1$-replica-symmetry-breaking (RSB) phase. For any $T_0=T$ above the dynamical phase transition point $T_c^{\rm dyn}$ the resulting stationary relaxation dynamics coincide with the FDT solution for these equations, while for lower $T_0=T$ in the $1$-RSB phase, the relaxation dynamics coincides with the FDT solution, now concentrated on the single spherical band within the Gibbs measure's support on which the initial point lies.

math.PR↗
arXiv2024

Disordered Gibbs measures and Gaussian conditioning

We study the law of a random field $f_N(\boldsymbol{\sigma})$ evaluated at a random sample from the Gibbs measure associated to a Gaussian field $H_N(\boldsymbol{\sigma})$. In the high-temperature regime, we show that bounds on the probability that $f_N(\boldsymbol{\sigma})\in A$ for $\boldsymbol{\sigma}$ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic $\boldsymbol{\sigma}$ under the conditional Gaussian law given that $H_N(\boldsymbol{\sigma})/N=E$ for $E$ close to the derivative $F'(\beta)$ of the free energy (which is the typical value of $H_N(\boldsymbol{\sigma})/N$ under the Gibbs measure). In the more challenging low-temperature regime we restrict to $k$-RSB spherical spin glasses, proving a similar result, now with a more elaborate conditioning. Namely, with $q_i$ denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that $H_N(\boldsymbol{\sigma})/N=E$, here one needs to also condition on the energy and its gradient at points $\mathbf{x}_1,\ldots,\mathbf{x}_k$ such that $\langle \mathbf{x}_i,\mathbf{x}_j\rangle/N=q_{i\wedge j}$ and $\langle \mathbf{x}_i,\boldsymbol{\sigma}\rangle/N\approx q_{i}$. Like in the high-temperature phase, the energy and gradient values on which one conditions are also specified by the model's Parisi measure. We apply our general results to two important problems from statistical physics. That is, computing the Franz-Parisi potential at any temperature and, reducing certain asymptotics of Langevin dynamics with initial conditions distributed according to the Gibbs measure, to the more manageable problem of studying dynamics with non-random initial conditions and conditional disorder.

math.PR↗
arXiv2024

On Smale's 17th problem over the reals

We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of arbitrary degrees. In the complex case and for $n= d-1$, Beltr\'{a}n and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe a polynomial time algorithm that finds solutions (with high probability) for $n= d -O(\sqrt{d\log d})$ if the maximal degree is bounded by $d^2$ and for $n=d-1$ if the maximal degree is larger than $d^2$.

cs.DS↗
arXiv2023

Solving systems of Random Equations via First and Second-Order Optimization Algorithms

We revisit the problem of solving $n$ random equations in $d$ real variables, when the equations are independent realizations of a Gaussian process in $d$ dimensions. A special case is the one of random polynomial equations, which has been studied since Littlewood-Offord and Kac in the 1940s (who studied of existence of solutions of random polynomials) and Shub and Smale in the 1990s. The last authors first investigated the computational aspect of this problem. Smale's `17th problem' asks whether a system of random polynomial equations can be (approximately) solved in average case polynomial time. We formulate this as a nonconvex optimization problem, and apply local algorithms based on gradient or Hessian information. We leverage recent advances in spin glass theory to characterize the optimal algorithm in this class, and show that the latter undergoes a phase transition at a critical value $\alpha_{\text{alg}}$ of the ratio $\alpha=n/d$. We establish that near-solutions can be found with-high probability for $\alpha<\alpha_{\text{alg}}$, while a companion paper proves that a broad class of efficient algorithms fail for $\alpha>\alpha_{\text{alg}}$ (we outline the proof of this hardness result). We further prove that there are cases such that for $(1+\delta)\alpha_{\text{alg}} 0$ arbitrarily small) solutions exists with high probability but are not found efficiently by a broad class of algorithms. We compare our predictions with numerical simulations using the optimal algorithm we propose as well as stochastic gradient descent, and show that they are accurate for a related albeit non-Gaussian cost function. We finally observe empirically a sensitivity cross-over in the behavior of optimization algorithms, below $\alpha_{\text{alg}}$. This marks a qualitative departure with respect to standard optimization theories.

math.PR↗
arXiv2023

Concentration for the zero set of large random polynomial systems

For random systems of $K$ polynomials in $N + 1$ real variables which include the models of Kostlan (1987) and Shub and Smale (1993), we prove that the number of zeros on the unit sphere for $K = N$ or the Hausdorff measure of the zero set for $K < N$ concentrates around its mean as $N\to\infty$. To prove concentration we show that the variance of the latter random variable normalized by its mean goes to zero. The polynomial systems we consider depend on a set of parameters which determine the variance of their Gaussian coefficients. We prove that the convergence is uniform in those parameters and $K$.

math.PR↗
arXiv2022

Optimization of random high-dimensional functions: Structure and algorithms

Replica symmetry breaking postulates that near optima of spin glass Hamiltonians have an ultrametric structure. Namely, near optima can be associated to leaves of a tree, and the Euclidean distance between them corresponds to the distance along this tree. We survey recent progress towards a rigorous proof of this picture in the context of mixed $p$-spin spin glass models. We focus in particular on the following topics: $(i)$~The structure of critical points of the Hamiltonian; $(ii)$~The realization of the ultrametric tree as near optima of a suitable TAP free energy; $(iii)$~The construction of efficient optimization algorithm that exploits this picture.

math.PR↗
arXiv2022

On-Site Potential Creates Complexity in Systems with Disordered Coupling

We calculate the average number of critical points $\overline{\mathcal{N}}$ of the energy landscape of a many-body system with disordered two-body interactions and a weak on-site potential. We find that introducing a weak nonlinear on-site potential dramatically increases $\overline{\mathcal{N}}$ to exponential in system size and give a complete picture of the organization of critical points. Our results extend solvable spin-glass models to physically more realistic models and are of relevance to glassy systems, nonlinear oscillator networks and many-body interacting systems.

cond-mat.dis-nn↗
arXiv2022

On the second moment method and RS phase of multi-species spherical spin glasses

Excluding some special cases, computing the critical inverse-temperature $β_c$ of a mixed $p$-spin spin glass model is a difficult task. The only known method to calculate its value for a general model requires the full power of the Parisi formula. On the other hand, an easy application of the second moment method to the partition function yields an explicit lower bound $β_m\leq β_c$ to the critical inverse-temperature. Interestingly, in the important case of the Sherrington-Kirkpatrick model $β_m=β_c$. In this work we consider the multi-species spherical mixed $p$-spin models without external field, and characterize by a simple condition the models for which the second moment method works in the whole replica symmetric phase, namely, models such that $β_m=β_c$. In particular, for those models we obtain the value of $β_c$.

math.PR↗
arXiv2022

TAP approach for multi-species spherical spin glasses II: the free energy of the pure models

In a companion paper we developed the generalized TAP approach for general multi-species spherical mixed $p$-spin models. In this paper, we use it to compute the limit of the free energy at any temperature for all pure multi-species spherical $p$-spin models, assuming that certain free energies converge. Importantly, the pure multi-species models do not satisfy the convexity assumption on the mixture which was crucial in the recent proofs of the Parisi formula for the multi-species Sherrington-Kirkpatrick model by Barra et al. (2015) and Panchenko (2015) and for the multi-species spherical mixed p-spin models by Bates and Sohn (2021).

math.PR↗
arXiv2022

Convergence of the free energy for spherical spin glasses

We prove that the free energy of any spherical mixed $p$-spin model converges as the dimension $N$ tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the well-known Guerra-Toninelli interpolation method. The latter was invented for models with Ising spins to prove that the free energy is super-additive and therefore (normalized by $N$) converges. In the spherical case, however, the configuration space is not a product space and the interpolation cannot be applied directly. We first relate the free energy on the sphere of dimension $N+M$ to a free energy defined on the product of spheres in dimensions $N$ and $M$ to which we then apply the interpolation method. This yields an approximate super-additivity which is sufficient to prove the convergence.

math.PR↗
arXiv2021

TAP approach for multi-species spherical spin glasses I: general theory

We develop a generalized TAP approach for the multi-species version of the spherical mixed $p$-spin models. In particular, we prove a generalized TAP representation for the free energy at any overlap vector which is multi-samplable in an appropriate sense. Moreover, we show that if a multi-samplable overlap is maximal, then the TAP correction is equal to an analogue of the well-known Onsager reaction term. Finally, in a companion paper we use the results from the current paper to compute the free energy at any temperature for all multi-species pure $p$-spin models, assuming the free energy converges.

math.PR↗
arXiv2021

Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies

We consider critical points of the spherical pure $p$-spin spin glass model with Hamiltonian $H_{N}\left(\boldsymbol{\sigma}\right)=\frac{1}{N^{\left(p-1\right)/2}}\sum_{i_{1},...,i_{p}=1}^{N}J_{i_{1},...,i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}}$, where $\boldsymbol{\sigma}=\left(\sigma_{1},...,\sigma_{N}\right)\in \mathbb{S}^{N-1}:=\left\{ \boldsymbol{\sigma}\in\mathbb{R}^{N}:\,\left\Vert \boldsymbol{\sigma}\right\Vert _{2}=\sqrt{N}\right\} $ and $J_{i_{1},...,i_{p}}$ are i.i.d standard normal variables. Using a second moment analysis, we prove that for $p\geq 32$ and any $E>-E_\infty$, where $E_\infty$ is the (normalized) ground state, the ratio of the number of critical points $\boldsymbol{\sigma}$ with $H_N(\boldsymbol{\sigma})\leq NE$ and its expectation asymptotically concentrates at $1$. This extends to arbitrary $E$ a similar conclusion of [Sub17a].

math.PR↗
arXiv2020

Dynamics for spherical spin glasses: disorder dependent initial conditions

We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed $p$-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure $p$-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).

math.PR↗
arXiv2019

Following the ground-states of full-RSB spherical spin glasses

We focus on spherical spin glasses whose Parisi distribution has support of the form $[0,q]$. For such models we construct paths from the origin to the sphere which consistently remain close to the ground-state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian. For finite mixtures $ν(x)$ it provides an algorithm of time complexity $O(N^{{\rm deg}(ν)})$ to find w.h.p. points with the ground-state energy, up to a small error. For the pure spherical models, the same algorithm reaches the energy $-E_{\infty}$, the conjectural terminal energy for gradient descent. Using the TAP formula for the free energy, for full-RSB models with support $[0,q]$, we are able to prove the correct lower bound on the free energy (namely, prove the lower bound from Parisi's formula), assuming the correctness of the Parisi formula only in the replica symmetric case.

math.PR↗
arXiv2019

The generalized TAP free energy II

In a recent paper [14], we developed the generalized TAP approach for mixed $p$-spin models with Ising spins at positive temperature. Here we extend these results in two directions. We find a simplified representation for the energy of the generalized TAP states in terms of the Parisi measure of the model and, in particular, show that the energy of all states at a given distance from the origin is the same. Furthermore, we prove the analogues of the positive temperature results at zero temperature, which concern the ground-state energy and the organization of ground-state configurations in space.

math.PR↗
arXiv2018

The generalized TAP free energy

We consider the mixed $p$-spin mean-field spin glass model with Ising spins and investigate its free energy in the spirit of the TAP approach, named after Thouless, Anderson, and Palmer. More precisely, we define and compute the generalized TAP correction, and establish the corresponding generalized TAP representation for the free energy. In connection with physicists' replica theory, we introduce the notion of generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameters match the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, our result shows that the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards-Anderson parameter.

math.PR↗