Estimation of the Euler method error on a Riemannian manifold

Bielecki, Andrzej

doi:10.1002/cnm.516

Cited by 9 publications

(7 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2) As noticed before, the forward Euler scheme does not guarantee X (n+1) ∈ M N even if X (n) ∈ M N in general. Thus, we take the update of x i (n) to the tangential direction f i (X (n)) via an exponential map [5,17]. Before we move on further, we briefly recall the exponential map below.…”

Section: Discretization Methodsmentioning

confidence: 99%

Emergent behaviors of discrete Lohe aggregation flows

Choi¹,

Ha²,

Park³

2021

Preprint

View full text Add to dashboard Cite

The swarm sphere (or Lohe sphere) model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the Lohe type aggregation models and study their emergent behaviors using Lyapunov function method. For the suitable discretization of the swarm sphere model, we use predictor-corrector method consisting of two steps. In the first step, we solve the first-order forward Euler scheme to get a predicted value, and in the second step, we project the predicted one into the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical sufficient frameworks leading to complete state aggregation and asymptotic state-locking.

show abstract

Section: Discretization Methodsmentioning

confidence: 99%

Emergent behaviors of discrete Lohe aggregation flows

Choi¹,

Ha²,

Park³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In this paper, we investigate invariance properties within the framework of Riemannian geometry and numerical differential equation solving. We observe that both the exact solution of the natural gradient dynamics and its approximation obtained with Riemannian Euler method (Bielecki, 2002) are invariant. We propose to measure the invariance of a numerical scheme by studying its rate of convergence to those idealized truly invariant solutions.…”

Section: Introductionmentioning

confidence: 90%

“…The Riemannian Euler method (see pp.3-6 in (Bielecki, 2002)) is a less common variant of the Euler method, which uses the Exponential map for its updates as x k+1 = Exp(x k , hf (t k , x k )), t k+1 = t k + h. While having the same asymptotic error O(h) as forward Euler, it has more desirable invariance properties.…”

Section: Numerical Differential Equation Solversmentioning

confidence: 99%

See 1 more Smart Citation

Accelerating Natural Gradient with Higher-Order Invariance

Song,

Ermon

2018

Preprint

View full text Add to dashboard Cite

An appealing property of the natural gradient is that it is invariant to arbitrary differentiable reparameterizations of the model. However, this invariance property requires infinitesimal steps and is lost in practical implementations with small but finite step sizes. In this paper, we study invariance properties from a combined perspective of Riemannian geometry and numerical differential equation solving. We define the order of invariance of a numerical method to be its convergence order to an invariant solution. We propose to use higher-order integrators and geodesic corrections to obtain more invariant optimization trajectories. We prove the numerical convergence properties of geodesic corrected updates and show that they can be as computational efficient as plain natural gradient. Experimentally, we demonstrate that invariance leads to faster optimization and our techniques improve on traditional natural gradient in deep neural network training and natural policy gradient for reinforcement learning.

show abstract

“…As noticed before, the forward Euler scheme does not guarantee X (n + 1) ∈ M N even if X (n) ∈ M N in general. Thus, we take the update of x i (n) to the tangential direction f i (X (n)) via an exponential map [5,17]. Before we move on further, we briefly recall the exponential map below.…”

Section: This Yields Minmentioning

confidence: 99%

Emergent behaviors of discrete Lohe aggregation flows

Choi

Park

2022

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.</p>

show abstract

Estimation of the Euler method error on a Riemannian manifold

Cited by 9 publications

References 24 publications

Emergent behaviors of discrete Lohe aggregation flows

Emergent behaviors of discrete Lohe aggregation flows

Accelerating Natural Gradient with Higher-Order Invariance

Emergent behaviors of discrete Lohe aggregation flows

Contact Info

Product

Resources

About