arXiv research

Lyapunov analysis improves RNN performance prediction.

Study approximates top Lyapunov exponents for surface mapping classes.

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…

On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of cu…

This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichmüller theory, and the spectral theory of one-frequency Schrödinger operators. The inspiration for…

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ 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…

Study on convergence of SDEs using entropy methods.

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…

Unified analysis of stochastic iterative algorithms using Lyapunov functions.

Lyapunov-based analysis shows polynomial sample complexity for WCMDPs and RBs.

New method stabilizes deep neural networks by setting Lyapunov exponent to zero.

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 …

New analysis improves SGD for robust and quantile regression with sub-quadratic convergence.

Lyapunov exponents help understand RNN stability.

We consider vector fields $X$ on a closed manifold $M$ with rest points of Morse type. For such vector fields we define the property of exponential growth. A cohomology class $ξ\in H^1(M;\mathbb R)$ which is Lyapunov for $X$ defines counting functions for isolated instantons and closed trajectories. If $X$ has exponent…

Unified framework for finite-sample RL algorithms using Lyapunov theory.

New insights into using momentum for non-convex optimization.

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

New methods accelerate gradient descent for convex and strongly convex functions.

New bounds show SGD can match deterministic gradient descent's convergence rate.

In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on $S^3$, we study Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$. More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1 …

This paper analyzes and improves monotonic accelerated algorithms like M-NAG and M-FISTA.

Optimal algorithms for Riemannian optimization with reduced complexity.

New approach to concentration inequalities for unbounded state space dynamical systems.

Refines geometric center of mass analysis for Einstein field equations.

We study the dynamical behaviors of degenerate stochastic differential equations (SDEs). We select an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conduct the Lyapunov exponential convergence analysis of degenerate SDEs. We derive the convergence rate cond…

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

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.…

Policy gradient algorithm with variable learning rates achieves near-optimal performance in multi-arm bandit problems.

Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.

Double Q-learning has the same mean-squared error as Q-learning under certain conditions.

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…

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…

The paper analyzes deep neural networks using control theory to set a time limit for their convergence.

New neural methods for stable control with provable guarantees.

Combines Lyapunov functions with controller synthesis for safe control policies.

The paper proves conditions for non-uniform expansion in partially hyperbolic systems.

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…

Paper analyzes SA for fixed-point equations with noise, establishing convergence rates.

We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …

Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.

We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmuell…

Proves simplicity of Lyapunov exponents for specific Anosov flows.

Study of deep neural networks using finite-time Lyapunov exponents.

Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof us…

Stabilizes complex systems using diffusion models trained on Lyapunov functions.

We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k exponents is greater or equal than the sum of the degree of any rank k holomorphic s…

We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…