Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is…
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48 results for “Lyapunov stability”
On Lyapunov Exponents for RNNs: Understanding Information Propagation Using Dynamical Systems Toolscs.LG
Lyapunov exponents help understand RNN stability.
problem Optimizing RNNs is sensitive to various parameters.
method Use Lyapunov exponents as dynamical system tools.
result Lyapunov spectrum measures training stability.
Stabilizes complex systems using diffusion models trained on Lyapunov functions.
problem Generating stabilizing controllers for complex dynamical systems.
method Trains a diffusion model on pairs of asymptotically stable vector fields and their Lyapunov functions to identify the closest stable field and adjust control functions.
result Efficient and rapid stabilization of unseen systems, showcasing generalizability.
New method stabilizes deep neural networks by setting Lyapunov exponent to zero.
problem Stability issues in deep neural networks with low width.
method Lyapunov initialization method to set Lyapunov exponent to zero.
result Lyapunov exponent governs stability of deep networks; standard methods fail for low width.
Neural Lyapunov Redesigneess.SY
Combines Lyapunov functions with controller synthesis for safe control policies.
problem Ensuring safety in controller design for nonlinear systems.
method Iterative algorithm combining Lyapunov function estimation and controller synthesis.
result Effective control policies with large safe regions are derived.
New neural methods for stable control with provable guarantees.
problem Designing stable control policies for nonlinear systems.
method Neural network Lyapunov functions and a falsifier to guide learning.
result Provable stability of controlled nonlinear systems.
Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimizationmath.OC
The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.
problem Minimizing nonsmooth nonconvex functions with convergence guarantees.
method Developed a framework for stochastic subgradient methods with global stability guarantees.
result Iterates are uniformly bounded and asymptotically stabilize around the stable set of the differential inclusion.
In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine…
New RL policy for unbounded state space with stability guarantee.
problem Traditional RL methods fail for unbounded state space.
method Proposes stability as performance metric, uses Sparse-Sampling-based Monte Carlo Oracle.
result Proposed policy ensures state dynamics remain bounded with high probability.
Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.
problem Gradient instability in RNNs leading to exploding and vanishing gradients.
method Regularizing Lyapunov exponents through backpropagation using differentiable linear algebra.
result Gradient flossing improves RNN training success rate and convergence speed.
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
Paper proves convergence of SA algorithm via martingale and converse Lyapunov methods.
problem Proves convergence of stochastic approximation algorithm.
method Uses martingale and converse Lyapunov methods to prove convergence.
result Provides alternate proof of convergence for SA algorithm.
AROS uses Lyapunov-stabilized embeddings to improve out-of-distribution detection robustness against adversarial attacks.
problem Robust out-of-distribution (OOD) detection against adversarial attacks.
method Neural Ordinary Differential Equations (NODEs) with Lyapunov stability theory for generating robust embeddings.
result Improves robust detection performance significantly, e.g., from 37.8% to 80.1% on CIFAR-10 vs. CIFAR-100.
Paper analyzes stability and forgetting in score-based generative models.
problem Understanding the stability and long-time behavior of generative models.
method Quantitative bounds on sampling error using stability and forgetting properties of the Markov chain.
result Provides practical consequences of stability and contraction mechanism in sampling.
KCRL learns stable policies for nonlinear systems with formal guarantees.
problem Lack of stabilization guarantees in RL methods for safety-critical systems.
method KCRL uses Krasovskii's Lyapunov functions as a stability constraint and a primal-dual approach to learn stabilizing policies.
result KCRL guarantees learning a stabilizing policy in a finite number of interactions.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
Stable deep models learn dynamical systems with formal stability guarantees.
problem Difficulties in making formal claims about stability of deep network dynamics models.
method Jointly learning a dynamics model and Lyapunov function to ensure non-expansiveness.
result Proposes an approach for stable deep learning of dynamical systems.
Paper presents neural network controllers for offset-free setpoint tracking.
problem Offset-free setpoint tracking using neural network controllers.
method Exploiting slope-restricted activation functions, linear matrix inequalities are used to verify stability.
result Global and local stability conditions for neural network controllers are derived.
While training error of most deep neural networks degrades as the depth of the network increases, residual networks appear to be an exception. We show that the main reason for this is the Lyapunov stability of the gradient descent algorithm: for an arbitrarily chosen step size, the equilibria of the gradient descent ar…
New stability theory for Sinkhorn semigroups with explicit decay rates.
problem Stability and convergence of Sinkhorn iterations for various divergences.
method Operator-theoretic framework based on Lyapunov techniques.
result Explicit exponential decay rates for Sinkhorn iterates.
In this paper, we consider the stochastic iterative counterpart of the value iteration scheme wherein only noisy and possibly biased approximations of the Bellman operator are available. We call this counterpart as the approximate value iteration (AVI) scheme. Neural networks are often used as function approximators, i…
The paper analyzes how disturbances affect the convergence of algorithms in complex systems.
problem Analyzing the impact of disturbances on algorithm convergence in complex systems.
method Leveraging converse Lyapunov theorems, the paper derives stability bounds and convergence rates in the presence of disturbances.
result Key inequalities quantify the impact of disturbances on algorithmic performance.
New model stabilizes asynchronous LTI systems, independent of synchronous stability.
problem Stability of asynchronous LTI systems under randomization and asynchrony.
method Introduced a new model for random asynchronous LTI systems and developed a method for system identification.
result Stability of random asynchronous LTI systems is independent of synchronous stability.
Deep neural networks (DNNs) are vulnerable to subtle adversarial perturbations applied to the input. These adversarial perturbations, though imperceptible, can easily mislead the DNN. In this work, we take a control theoretic approach to the problem of robustness in DNNs. We treat each individual layer of the DNN as a …
Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.
problem Convergence analysis of population-based optimization methods
method Introduce an operator calculus for describing composite mean-field algorithms as compositions of elementary operators acting on probability measures.
result Establish a modular Lyapunov principle for certifying exponential decay of state-space Lyapunov function and search errors.
This paper uses dynamical systems to analyze and ensure convergence of the Bayesian EM algorithm.
problem Ensuring convergence of the Bayesian EM algorithm in incomplete-data scenarios.
method Applying Lyapunov stability theory to discrete-time dynamical systems.
result Conditions for convergence and potential for fast convergence of MAP-EM are established.
We consider the heat flow of corotational harmonic maps from to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…
Policy gradient algorithm with variable learning rates achieves near-optimal performance in multi-arm bandit problems.
problem Optimizing a policy gradient algorithm for multi-arm bandit problems with variable learning rates.
method Applied Foster-Lyapunov techniques to analyze a Markov chain formed by the state of the algorithm.
result The policy gradient algorithm converges to the optimal arm with logarithmic or poly-logarithmic regret.
A ML model accurately replicates chaotic dynamics across various parameters.
problem Replicating chaotic characteristics of non-linear dynamics using machine learning.
method A ML model trained to predict one-step-ahead states from historic states captures bifurcation diagrams and Lyapunov exponents universally.
result Variational quantum circuit outperforms classical models in reproducing long-term chaotic characteristics.
The paper develops techniques to study dynamical systems with Carnot metrics.
problem Understanding smooth dynamical systems in the presence of Carnot metrics.
method Employing techniques from Margulis-Mostow, Métivier, Mitchell, and Pansu on tangent cones, the paper establishes resonances between Lyapunov exponents.
result Local rigidity properties of higher hyperbolic rank metrics and uniform lattice actions on quaternionic and octonionic symmetric spaces.
Unified framework for solving fixed-point equations in deterministic and stochastic settings.
problem Solving fixed-point equations for seminorm-contractive operators in both deterministic and stochastic contexts.
method Fixed-point theorem and stochastic approximation analysis.
result Unified finite-sample bounds for various reinforcement learning algorithms.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
The paper studies random dynamical systems of polynomial automorphisms on C^2 and finds mean stability.
problem Random dynamical systems of polynomial automorphisms on C^2.
method Generic random dynamical systems of polynomial automorphisms are shown to have mean stability.
result A generic random dynamical system of polynomial automorphisms on C^2 has mean stability.
Develops a new theory for neural systems stability and width effects.
problem Stability and finite-width effects in deep neural systems.
method Gauge-covariant stochastic effective field theory using classical commuting fields.
result Predicts the edge of chaos and low-frequency spectral deformation.
In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on , we study Lyapunov graphs of nonsingular Smale flows on . More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1 …
RELTA-SGLD: Relative-Growth Localized Taming for Nonconvex Stochastic-Gradient Langevin Learningstat.ML
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
Proposes a method to learn system dynamics and region of attraction from trajectories.
problem Learning accurate dynamics and region of attraction from system trajectories.
method Uses local stability information as a prior to learn vector field and region of attraction.
result Efficient sampling and accurate estimate of dynamics in inner approximation of region of attraction.
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family.…
Lyapunov analysis improves RNN performance prediction.
problem Uncertainty in RNN performance prediction due to hyperparameters and architecture.
method Lyapunov spectral analysis of RNNs and Autoencoder-Lyapunov Embedding Learning (AeLLE).
result AeLLE successfully correlates RNN Lyapunov spectrum with accuracy and predicts performance.
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We a…
Reinforcement learning is a powerful paradigm for learning optimal policies from experimental data. However, to find optimal policies, most reinforcement learning algorithms explore all possible actions, which may be harmful for real-world systems. As a consequence, learning algorithms are rarely applied on safety-crit…
In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curv…
Equivalence of convex optimization, saddle-point problems, and variational inequalities is a well-established concept. The variational inequality (VI) is a static problem which is studied under dynamical settings using a framework called the projected dynamical system, whose stationary points coincide with the static s…
Study on financial systems using perturbed unimodal maps with heteroscedastic noise.
problem Analyzing systemic risk in financial systems using mathematical models.
method Investigation of one-dimensional unimodal maps perturbed by heteroscedastic noise, proving stability, convergence, and Lyapunov exponent continuity.
result Continuous dependence of average Lyapunov exponent on Markov chain parameters, and Gumbel's law for extreme values.
We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors. As a corollary, we complete the classification of commensurability…
On positive Lyapunov exponents along and non-uniformly expanding for partially hyperbolic systemsmath.DS
The paper proves conditions for non-uniform expansion in partially hyperbolic systems.
problem Conditions for non-uniform expansion in partially hyperbolic systems.
method Analysis of Lyapunov exponents and dominated splittings.
result Existence of physical SRB measure under specific conditions.
This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analyti…