Exploring high-dimensional random landscapes: from spin glasses to random matrices, passing through simple chaotic systems

TL;DR

The work develops a unified framework linking high-dimensional random landscapes to dynamics, centering on the Kac-Rice method for counting equilibria and dynamical mean-field theory (DMFT) to capture long-time behavior. It first clarifies the landscape methods for gradient and non-gradient systems, applying these tools to the pure spherical $p$-spin as a prototypical glassy model and to non-reciprocal neural-network-like settings. It then delivers three core advances: (i) exact dynamic-static comparisons in solvable non-reciprocal models, (ii) a mean-field computation of annealed complexity for the Sompolinsky–Crisanti–Sommers (SCS) neural network across non-reciprocity, and (iii) curvature-based and triplet-based analyses of barriers and local minima in the $p$-spin landscape, complemented by results on overlaps of eigenvectors in spiked random matrices. Collectively, the results sharpen understanding of how landscape structure governs dynamics in high-dimensional systems and introduce tools to bridge statics and dynamics in both gradient and non-gradient contexts, with implications for glassy physics, neural networks, and random-mitness landscapes. The work also broadens the landscape toolbox by connecting two-point and three-point complexities with curvature-driven pathways and by detailing spectral properties of conditioned Hessians in spiked matrix settings. This has potential applications in optimization, inference, and understanding activation dynamics in complex many-body systems.

Abstract

High-dimensional random landscapes underlie phenomena as diverse as glassy physics and optimization in machine learning, and even their simplest toy models already display extraordinarily rich behavior. This thesis aims to deepen our understanding of that behavior, by combining landscape-based approaches, via the Kac-Rice formalism, with dynamical approaches, paying special attention to both systems with reciprocal and with non-reciprocal interactions. After surveying core techniques and results through the spherical p-spin model, this thesis delivers three main advances: (i) exact dynamic-static comparison in a solvable class of models with non-reciprocal interactions, pinpointing differences and similarities of the two approaches; (ii) a stability-based calculation of the mean number of fixed points (i.e., annealed complexity) of the Sompolinsky-Crisanti-Sommers random neural network, for any level of non-reciprocity; (iii) two approaches to probe the barriers and the distribution of deep local minima in the landscape of the p-spin model; (iv) some results on the overlaps among eigenvectors of spiked, correlated random matrices, which are useful to explore the geometry of energy landscapes. Together, these results sharpen our understanding of these systems, while providing new tools and opening new doors for future research directions.

Figures (54)

• Figure 1: Comparison of $E_N$ with numerical simulations of the number of real roots of random polynomials $p_N(x)$. For each $N$ we sample $4000$ different polynomials and average their number of real roots (found with Python's method numpy.root).
• Figure 2: Idealized representation of the spectral density of the matrix $\nabla^2_\perp h$ for $N\to\infty$. From left to right the energy density $\epsilon$ is increased, showing a transition in the stability of the Hessian, from local minima (positive spectrum) to marginal minima (the left edge touches zero) to saddles (a portion of the spectrum is negative). The spectrum is always centered at $p\epsilon\sqrt{2}$.
• Figure 3: Plot of the complexity $\Sigma(\epsilon)$ for the case $p=3$. To the right of $\epsilon_{th}$ the complexity refers to saddle points with an extensive number of negative eigenvalues of the Hessian. At $\epsilon_{th}$ the stationary points are marginal minima, see Fig. \ref{['fig:stability_pspin_fp']}, whereas between $\epsilon_{gs}$ and $\epsilon_{th}$ they are local minima. At $\epsilon_{gs}$ they are absolute minima.
• Figure 4: Numerical solution of the equation for $C_{TTI}(\tau)$ in the TTI regime, for a range of energies above (red) and below (green) $T_d$. The $x$-axis is in log scale. We used Euler discretization with a time step $dt = 0.01$ and $p=3$.
• Figure 5: Plot of $g(q)$ for various temperatures, showing that at $T_d$ (orange curve) the second minimum touches the $q$-axis.

Theorems & Definitions (2)

• Lemma 1
• proof