Tootfinder

Learning Neural Networks by Neuron Pursuit
Akshay Kumar, Jarvis Haupt
https://arxiv.org/abs/2509.12154 https://arxiv.org/pdf/2509.12154

Learning Neural Networks by Neuron Pursuit
The first part of this paper studies the evolution of gradient flow for homogeneous neural networks near a class of saddle points exhibiting a sparsity structure. The choice of these saddle points is motivated from previous works on homogeneous networks, which identified the first saddle point encountered by gradient flow after escaping the origin. It is shown here that, when initialized sufficiently close to such saddle points, gradient flow remains near the saddle point for a sufficiently lon…

Zero-sum turn games using Q-learning: finite computation with security guarantees
Sean Anderson, Chris Darken, Jo\~ao Hespanha
https://arxiv.org/abs/2509.13585 https://

Zero-sum turn games using Q-learning: finite computation with security guarantees
This paper addresses zero-sum ``turn'' games, in which only one player can make decisions at each state. We show that pure saddle-point state-feedback policies for turn games can be constructed from dynamic programming fixed-point equations for a single value function or Q-function. These fixed-points can be constructed using a suitable form of Q-learning. For discounted costs, convergence of this form of Q-learning can be established using classical techniques. For undiscounted costs, we provi…

Randomized coordinate gradient descent almost surely escapes strict saddle points
Ziang Chen, Yingzhou Li, Zihao Li
https://arxiv.org/abs/2508.07535 https://

Randomized coordinate gradient descent almost surely escapes strict saddle points
We analyze the behavior of randomized coordinate gradient descent for nonconvex optimization, proving that under standard assumptions, the iterates almost surely escape strict saddle points. By formulating the method as a nonlinear random dynamical system and characterizing neighborhoods of critical points, we establish this result through the center-stable manifold theorem.

Testing the effective action approach to bubble nucleation in holography
Oscar Henriksson, Niko Jokela, Xin Li
https://arxiv.org/abs/2507.11622 https://

Testing the effective action approach to bubble nucleation in holography
The nucleation of bubbles during a first-order phase transition has recently been explored using holographic duality, which can provide an important complement to standard perturbative methods. These computations typically require finding static and spatially inhomogeneous saddle points, known as critical bubbles, which correspond in the gravitational dual to solutions of nonlinear partial differential equations. A computationally simpler alternative is to use the gravitational dual to derive t…

Saddle-node bifurcation during the relaminarization of turbulent puffs in pipe
Basheer Ahmad Khan, Shai Arogeti, Oriel Shoshani, Alexander Yakhot
https://arxiv.org/abs/2508.08862

Saddle-node bifurcation during the relaminarization of turbulent puffs in pipe
Turbulent puffs in a pipe persist for a long time before abruptly transitioning to laminar flow through viscous exponential decay. Direct numerical simulation results reveal a saddle-node bifurcation sequence governing the final relaminarization stage. Specifically, a universal square-root scaling law in the annihilation of critical points that is indicative of a saddle-node bifurcation. The Riccati equation has been proposed as a straightforward approach to modeling these bifurcations.

Saddle Point Configurations for Spherical Ferromagnets
Stephen Gustafson, Daniel Meinert, Christof Melcher
https://arxiv.org/abs/2509.05159 https://arxiv.o…

Saddle Point Configurations for Spherical Ferromagnets
We investigate saddle point configurations in spherical ferromagnets with perpendicular anisotropy. These are modeled by a micromagnetic energy functional on the unit sphere that leads to the emergence of the so-called curvature induced Dzyaloshinskii--Moriya interaction. For this functional we establish the existence of two distinct types of saddle points with zero mapping degree. We use a parabolic flow approach inspired by the harmonic map heat flow where for certain highly symmetric initial…

Saddle-point structure of fixed points in a reaction-diffusion equation with discontinuous nonlinearity
Jos\'e Valero
https://arxiv.org/abs/2507.22452 https://

Saddle-point structure of fixed points in a reaction-diffusion equation with discontinuous nonlinearity
In this paper, we study the local behaviour of solutions near the fixed points of a reaction-diffusion equation with discontinuous nonlinearity. By employing an appropriate linearization around the fixed points, which involves the Dirac delta distribution, we analyze the stability of the stationary solutions and demonstrate that they exhibit a saddle-point structure. As a result, we establish the hyperbolicity of the fixed points.

A Brief Note on Complex AdS-Schwarzschild Black Holes
Raghu Mahajan, Kaustubh Singhi
https://arxiv.org/abs/2509.08883 https://arxiv.org/pdf/2509.08883

A Brief Note on Complex AdS-Schwarzschild Black Holes
In the context of thermodynamics of asymptotically anti-de Sitter spaces, it is often stated that at very low temperatures, there is only one saddle point available-namely, thermal AdS-and hence this sole saddle dictates the low-temperature behavior. However, AdS-Schwarzschild black holes continue to exist at low temperatures as complex saddle points. We point out that the real part of the on-shell action of these complex black holes is smaller than that of thermal AdS at the lowest temperature…

On the Convergence of a Noisy Gradient Method for Non-convex Distributed Resource Allocation: Saddle Point Escape
Lei Qin, Ye Pu
https://arxiv.org/abs/2508.06922 https://…

On the Convergence of a Noisy Gradient Method for Non-convex Distributed Resource Allocation: Saddle Point Escape
This paper considers a class of distributed resource allocation problems where each agent privately holds a smooth, potentially non-convex local objective, subject to a globally coupled equality constraint. Built upon the existing method, Laplacian-weighted Gradient Descent, we propose to add random perturbations to the gradient iteration to enable efficient escape from saddle points and achieve second-order convergence guarantees. We show that, with a sufficiently small fixed step size, the it…

Critical Point Processes Obtained from a Gaussian Random Field with a View Towards Statistics
Julien Chevallier (SVH, LJK), Jean-Fran\c{c}ois Coeurjolly (LJK, SVH), Rasmus Waagepetersen
https://arxiv.org/abs/2507.04753

Critical Point Processes Obtained from a Gaussian Random Field with a View Towards Statistics
This paper establishes the theoretical foundation for statistical applications of an intriguing new type of spatial point processes called critical point processes. These point processes, residing in Euclidean space, consist of the critical points of latent smooth Gaussian random fields or of subsets of critical points like minima, saddle points etc. Despite of the simplicity of their definition, the mathematical analysis of critical point processes is non-trivial involving for example deep res…

Escaping Saddle Points via Curvature-Calibrated Perturbations: A Complete Analysis with Explicit Constants and Empirical Validation
Faruk Alpay, Hamdi Alakkad
https://arxiv.org/abs/2508.16540

Escaping Saddle Points via Curvature-Calibrated Perturbations: A Complete Analysis with Explicit Constants and Empirical Validation
We present a comprehensive theoretical analysis of first-order methods for escaping strict saddle points in smooth non-convex optimization. Our main contribution is a Perturbed Saddle-escape Descent (PSD) algorithm with fully explicit constants and a rigorous separation between gradient-descent and saddle-escape phases. For a function $f:\mathbb{R}^d\to\mathbb{R}$ with $\ell$-Lipschitz gradient and $ρ$-Lipschitz Hessian, we prove that PSD finds an $(ε,\sqrt{ρε})$-approximate second-order st…

Conditions for Suppression of Gas Phase Chemical Reactions inside a Dark Infrared Cavity: O$_{2}$ 2NO$\to$ 2NO$_{2}$ as an example
Mwdansar Banuary, Ashish Kumar Gupta, Nimrod Moiseyev
https://arxiv.org/abs/2509.03299

Conditions for Suppression of Gas Phase Chemical Reactions inside a Dark Infrared Cavity: O$_{2}$ + 2NO$\to$ 2NO$_{2}$ as an example
The ability to slow down chemical reactions using a seemingly simple setup reactions confined within a cavity formed by two parallel mirrors is fascinating. However, theory and experiment have not yet fully converged. In this work, we provide the conditions and guidelines for selecting reactions that can be suppressed in a dark cavity. The primary requirement is that the reaction's potential energy surface must contain two saddle points (and not one saddle point as required for enhancement). Th…

Discrete Dynamics of a Phytoplankton-Zooplankton Model with Toxin-Mediated Interactions
Sobirjon Shoyimardonov
https://arxiv.org/abs/2508.07942 https://arx…

Discrete Dynamics of a Phytoplankton-Zooplankton Model with Toxin-Mediated Interactions
We investigate the dynamics of a discrete phytoplankton-zooplankton model incorporating Holling type~III predation and Holling type~II toxin release. The existence and stability of positive fixed points are analyzed, and it is shown that when two such points, $E_1$ and $E_2$, exist, $E_2$ is always a saddle. A Neimark-Sacker bifurcation at $E_1$ is verified using the normal form method, indicating the emergence of closed invariant curves. This bifurcation implies that phytoplankton and zooplank…

Second-order methods for provably escaping strict saddle points in composite nonconvex and nonsmooth optimization
Alexander Bodard, Masoud Ahookhosh, Panagiotis Patrinos
https://arxiv.org/abs/2506.22332

Second-order methods for provably escaping strict saddle points in composite nonconvex and nonsmooth optimization
This study introduces two second-order methods designed to provably avoid saddle points in composite nonconvex optimization problems: (i) a nonsmooth trust-region method and (ii) a curvilinear linesearch method. These developments are grounded in the forward-backward envelope (FBE), for which we analyze the local second-order differentiability around critical points and establish a novel equivalence between its second-order stationary points and those of the original objective. We show that the…

First-principles study of Rh- and Pd-based kagome-layered shandites
Luca Buiarelli, Turan Birol, Brian M. Andersen, Morten H. Christensen
https://arxiv.org/abs/2507.23329 https:…

First-principles study of Rh- and Pd-based kagome-layered shandites
The shandite structure hosts transition metals arranged in kagome layers stacked rhombohedrally, and interspersed with post-transition metal ions and chalcogens. The electronic states near the Fermi level are dominated by the transition metal $d$-orbitals and feature saddle points near several of the high-symmetry positions of the Brillouin zone, most notably the F and L points. Combining symmetry considerations with ab initio methods, we study the electronic and structural properties of these …

Decentralized Local Voltage Control for Active Distribution Networks
Diana Vieira Fernandes, Soummya Kar, Carlos Santos Silva
https://arxiv.org/abs/2509.09048 https://

Decentralized Local Voltage Control for Active Distribution Networks
Distribution networks face challenges from the increasing deployment of Distributed Energy Resources (DERs) and the emergence of bidirectional power flows. We propose a decentralized Volt/VAr control method based on a saddle-point reformulation and consensus+innovation (C+I) updates. Each agent at a controllable bus computes and enforces its own set-points using only neighbor communication. Our method embeds passive buses directly, preserves network physics through a linearized Jacobian model, …

Generic conservative dynamics on Stein manifolds with the volume density property
Leandro Arosio, Finnur Larusson
https://arxiv.org/abs/2507.23133 https://…

Generic conservative dynamics on Stein manifolds with the volume density property
We study the dynamics of generic volume-preserving automorphisms $f$ of a Stein manifold $X$ of dimension at least 2 with the volume density property. Among such $X$ are all connected linear algebraic groups (except $\mathbb{C}$ and $\mathbb{C}^*$) with a left- or right-invariant Haar form. We show that a generic $f$ is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in $X$. We present analogous results with …

Disorder-aided Early Warning Signals: Predicting Catastrophic Shifts in Athermal Systems
Tapas Bar, Anurag Banerjee, Blai Casals, Gustau Catalan, Javier Rodr\'iguez-Viejo
https://arxiv.org/abs/2509.01601

Disorder-aided Early Warning Signals: Predicting Catastrophic Shifts in Athermal Systems
The early prediction of tipping points, distinguished by sudden and catastrophic shifts from stable states, poses a challenging task that would enable us to assess the impending threat across natural and engineered systems. This threat becomes particularly acute in low-fluctuation environments, where tipping occurs through saddle-node bifurcation without prior warning in noise dynamics. In this study, we investigate the tipping point dynamics of avalanche catastrophes in low-fluctuation domain,…

Exact penalty functions and global saddle points of augmented Lagrangians for well-posed constrained optimization problems
M. V. Dolgopolik
https://arxiv.org/abs/2508.16462 http…

Exact penalty functions and global saddle points of augmented Lagrangians for well-posed constrained optimization problems
The goal of this article is to study necessary and sufficient conditions for the exactness of penalty functions and the existence of global saddle points of augmented Lagrangians for well-posed (in a suitable sense) constrained optimization problems in infinite dimensional spaces. To this end, we present a new version of extended well-posedness of a constrained optimization problem and analyse how it relates to the more well-known types of well-posedness, such as Tykhonov and Levitin-Polyak wel…

On gradient descent-ascent flows in metric spaces
Noboru Isobe, Sho Shimoyama
https://arxiv.org/abs/2506.20258 https://arxiv.org/pdf/…

On gradient descent-ascent flows in metric spaces
Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via maximal monotone operator theory, their extension to general metric spaces, particularly Wasserstein spaces, has remained largely unexplored. In this paper, we develop a mathematical theory of GDA flows on the product of two complete metric spaces, formulating…

Morse Index Classification and Landscape of Kuramoto System for Hebbian-based Binary Pattern Recognition
Xiaoxue Zhao, Xiang Zhou
https://arxiv.org/abs/2508.21310 https://

Morse Index Classification and Landscape of Kuramoto System for Hebbian-based Binary Pattern Recognition
This study examines the Kuramoto model with a Hebbian learning rule and second-order Fourier coupling for binary pattern recognition. The system stores memorized binary patterns as stable critical points, enabling it to identify the closest match to a defective input. However, the system exhibits multiple stable states and thus the dynamics are influenced by saddle points and other unstable critical points, which may disrupt convergence and recognition accuracy. We systematically classify the s…

Power System Voltage Stability Boundary: Computational Results and Applications
Zhenyao Li, Yifan Yao, Deqiang Gan
https://arxiv.org/abs/2508.03119 https://

Power System Voltage Stability Boundary: Computational Results and Applications
The objective of this paper is to report some computational results for the theory of DAE stability boundary, with the aim of advancing applications in power system voltage stability studies. Firstly, a new regularization transformation for standard differential-algebraic equations (DAEs) is proposed. Then the existence of anchor points on voltage stability boundary is examined, and an optimization method for computing the controlling pseudo-saddle is suggested. Subsequently, a local representa…

A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization
Po Chen, Rujun Jiang, Peng Wang
https://arxiv.org/abs/2506.20344 https://

A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization
Despite its wide range of applications across various domains, the optimization foundations of deep matrix factorization (DMF) remain largely open. In this work, we aim to fill this gap by conducting a comprehensive study of the loss landscape of the regularized DMF problem. Toward this goal, we first provide a closed-form expression of all critical points. Building on this, we establish precise conditions under which a critical point is a local minimizer, a global minimizer, a strict saddle po…

Gradient descent avoids strict saddles with a simple line-search method too
Andreea-Alexandra Mu\c{s}at, Nicolas Boumal
https://arxiv.org/abs/2507.13804 ht…

Gradient descent avoids strict saddles with a simple line-search method too
It is known that gradient descent (GD) on a $C^2$ cost function generically avoids strict saddle points when using a small, constant step size. However, no such guarantee existed for GD with a line-search method. We provide one for a modified version of the standard Armijo backtracking method with generic, arbitrarily large initial step size. In contrast to previous works, our analysis does not require a globally Lipschitz gradient. We extend this to the Riemannian setting (RGD), assuming the…