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.
Lyapunov exponents help understand RNN stability.
New method stabilizes deep neural networks by setting Lyapunov exponent to zero.
The paper refines optimization algorithms using Lyapunov functions and differential equations.
New algorithm reduces regret and constraint violation in constrained bandit problems.
Lyapunov-based analysis shows polynomial sample complexity for WCMDPs and RBs.
Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.
The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.
New bounds show SGD can match deterministic gradient descent's convergence rate.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
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…
Investigates optimal consumption and investment strategies with constraints in incomplete markets.
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…
New methods accelerate gradient descent for convex and strongly convex functions.
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.
Optimal algorithms for Riemannian optimization with reduced complexity.
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 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 insights into using momentum for non-convex optimization.
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…
New algorithm reduces communication time in federated learning.
The paper accelerates ISTA and FISTA algorithms for composite optimization problems.
New neural methods for stable control with provable guarantees.
A new RL model ensures safe learning in uncertain environments.
Combines Lyapunov functions with controller synthesis for safe control policies.
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…
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…
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.
Lion optimizer performs well in training AI models with memory efficiency.
New RL policy for unbounded state space with stability guarantee.
Study on convergence of SDEs using entropy methods.
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.
Policy gradient algorithm with variable learning rates achieves near-optimal performance in multi-arm bandit problems.
We study continuous action reinforcement learning problems in which it is crucial that the agent interacts with the environment only through safe policies, i.e.,~policies that do not take the agent to undesirable situations. We formulate these problems as constrained Markov decision processes (CMDPs) and present safe p…
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.
This paper analyzes and improves monotonic accelerated algorithms like M-NAG and M-FISTA.
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…
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…
RNNs struggle with chaotic dynamics due to exploding gradients, but we found a way to optimize training.