Triviality of the Geometry of Mixed p-Spin Spherical Hamiltonians with External Field

Belius, David; Černý, Jǐŕı; Nakajima, Shuta; Schmidt, Marius A.

doi:10.1007/s10955-021-02855-6

Cited by 14 publications

(5 citation statements)

References 21 publications

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“…Note that this special value e * is nothing else but the typical minimal cost e min from (32). It minimizes the function L(e) in (33), and moreover L(e * ) = 0 as expected and can easily checked from (33) by substituting there the values e * and v * .…”

Section: A Summary and Discussion Of The Main Resultsmentioning

confidence: 56%

“…Its actual range of validity is however not clear to us at the moment, and clarifying it remains an unsolved issue. To this end it is also worth mentioning that many questions for a simpler optimization problem on the sphere studied originally in [19] by a combination of rigorous Kac-Rice and heuristic replica approaches were subsequently successfully put on the firm mathematical ground in the series of papers [29][30][31][32][33]. We hope some of those techniques can be also useful in the context of present model as well.…”

Section: Discussionmentioning

confidence: 86%

“…As such, they call for a proper justification by mathematically rigorous methods. In particular, as long as σ > 0 the formula ( 33) is expected to be valid for some range of around the mean/typical value, that is also for some values e > e * , and indeed it is easy to plot it using (33), see the figures 3 and 4. Its actual range of validity is however not clear to us at the moment, and clarifying it remains an unsolved issue.…”

Section: Discussionmentioning

confidence: 98%

“…following from (110). In this way one completely solves the problem of explicitly inverting the Legendre transform and providing the required large deviation rate function (33) for the minimal cost as a function of e.…”

Section: Derivation Of the Minimal Cost And Its Large Deviation Funct...mentioning

confidence: 99%

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Optimization landscape in the simplest constrained random least-square problem

Fyodorov

Tublin

2022

J. Phys. A: Math. Theor.

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We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of $M$ linear equations in $N$ unknowns: $(\bm a_k,\bm x)=b_k, \, k=1,\ldots,M$ on the $N-$sphere ${\bf x}^2=N$. We treat both the $N-$component vectors $\bm a_k$ and parameters $b_k$ as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case $M>N$ in the framework of the Kac-Rice approach combined with the Random Matrix Theory for Wishart Ensemble. Then we perform its asymptotic analysis as $N\to \infty$ at a fixed $\alpha=M/N>1$ in various regimes. In particular, this analysis allows to extract the Large Deviation Function for the density of the smallest Lagrange multiplier $\lambda_{min}$ associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value ${\cal E}_{min}$ of the loss function as $N\to \infty$. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the Large Deviation function for the density of ${\cal E}_{min}$ at $N\gg 1$ and {\it any} fixed ratio $\alpha=M/N>0$. As a by-product, we find the {\it compatibility threshold} $\alpha_c<1$ which is the value of $\alpha$ beyond which a large random linear system on the $N-$sphere becomes typically incompatible.

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Section: A Summary and Discussion Of The Main Resultsmentioning

confidence: 56%

Section: Discussionmentioning

confidence: 86%

Section: Discussionmentioning

confidence: 98%

Section: Derivation Of the Minimal Cost And Its Large Deviation Funct...mentioning

confidence: 99%

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Optimization landscape in the simplest constrained random least-square problem

Fyodorov

Tublin

2022

J. Phys. A: Math. Theor.

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“…Intricate questions, such as the number of critical points with fixed index at given overlap from a minimum, are considered for pure 𝑝-spin models in [53]. We also mention [24] for an upper bound on the number of critical points of the TAP free energy of the Sherrington-Kirkpatrick model, and the recent works [12,13] on neural networks, [8] on Gaussian fields with isotropic increments, [16,29] on stable/unstable equilibria in systems of non-linear differential equations, and [14,27,30] on the phenomenon of "topological trivialization" for spherical spin glasses with an external field. In most of these models, the conditioned Hessian is closely related to the Gaussian Orthogonal Ensemble (GOE), a consequence of distributional symmetries of the landscapes.…”

Section: Rotationally Invariant Modelsmentioning

confidence: 99%

Landscape complexity beyond invariance and the elastic manifold

Ben Arous,

Bourgade,

McKenna

2023

Comm Pure Appl Math

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This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self‐interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal‐to‐noise model of soft spins in an anisotropic well, for which we prove a negative‐second‐moment threshold distinguishing positive from zero complexity. A universal near‐critical behavior appears within this phase portrait, namely quadratic near‐critical vanishing of the complexity of total critical points, and cubic near‐critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non‐invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

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The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses

Sellke

2024

Comm Pure Appl Math

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We study the Langevin dynamics for spherical ‐spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder‐dependent initialization and on exponential time‐scales.

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Triviality of the Geometry of Mixed p-Spin Spherical Hamiltonians with External Field

Cited by 14 publications

References 21 publications

Optimization landscape in the simplest constrained random least-square problem

Optimization landscape in the simplest constrained random least-square problem

Landscape complexity beyond invariance and the elastic manifold

The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses

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