A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New neural methods for stable control with provable guarantees.
New method stabilizes deep neural networks by setting Lyapunov exponent to zero.
Study on convergence of SDEs using entropy methods.
Lyapunov analysis improves RNN performance prediction.
Combines Lyapunov functions with controller synthesis for safe control policies.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
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 …
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…
Paper proves convergence of SA algorithm via martingale and converse Lyapunov methods.
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.…
New methods accelerate gradient descent for convex and strongly convex functions.
Neural network outperforms traditional methods in chaotic dynamics classification.
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of Schwartzman, whereas the methods are adapted from work of Sullivan.
Study approximates top Lyapunov exponents for surface mapping classes.
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 guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.
In many real-world reinforcement learning (RL) problems, besides optimizing the main objective function, an agent must concurrently avoid violating a number of constraints. In particular, besides optimizing performance it is crucial to guarantee the safety of an agent during training as well as deployment (e.g. a robot…
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…
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…
Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.
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.
Lyapunov exponents help understand RNN stability.
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.
Graphical models for covariance matrices improve structure learning.
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.
AROS uses Lyapunov-stabilized embeddings to improve out-of-distribution detection robustness against adversarial attacks.
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…
This paper analyzes and improves monotonic accelerated algorithms like M-NAG and M-FISTA.
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…
New analysis improves SGD for robust and quantile regression with sub-quadratic convergence.
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 findings on diffusion rates in wind-tree model with rational parameters.
Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.
Unified analysis of stochastic iterative algorithms using Lyapunov functions.
We construct a Teichmueller curve uniformized by the Fuchsian triangle group (m,n,\infty) for every m<n. Our construction includes the Teichmueller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find Bill…
We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.
Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.
New RL policy for unbounded state space with stability guarantee.
Study the topology of stable vector fields and Lyapunov functions on R^n.
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
Study describes splitting and filtration of Hodge bundle on quadratic differentials.