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Inexact Adaptive Cubic Regularization Algorithms for Optimization on Riemannian Manifolds and Their Application to Joint Diagonalization


Core Concepts
The paper proposes an inexact adaptive cubic regularization algorithm for solving large-scale separable unconstrained optimization problems on general Riemannian manifolds, which employs the inexact gradient and Hessian information. The algorithm is proven to have an iteration complexity of O(max{ε^-2_g, ε^-3_H}) for achieving the (ε_g, ε_H)-optimality under certain assumptions. The algorithm is applied to solve the joint diagonalization problem on the Stiefel manifold, and numerical experiments show that the inexact algorithms outperform the deterministic algorithm and the inexact trust-region algorithm.
Abstract
The paper proposes an inexact adaptive cubic regularization algorithm for solving large-scale separable unconstrained optimization problems on general Riemannian manifolds. The key highlights are: The algorithm employs the inexact gradient and Hessian information, which is particularly useful in large-scale settings where the exact gradient and Hessian computations can be expensive. The algorithm is proven to have an iteration complexity of O(max{ε^-2_g, ε^-3_H}) for achieving the (ε_g, ε_H)-optimality under certain assumptions on the accuracies of the inexact gradient and Hessian. The paper also establishes the iteration complexities of the deterministic Riemannian adaptive cubic regularization algorithm and the inexact Riemannian adaptive cubic regularization algorithm under the true gradient. As an application, the proposed algorithms are applied to solve the joint diagonalization problem on the Stiefel manifold. Numerical experiments show that the inexact algorithms outperform the deterministic algorithm and the inexact trust-region algorithm.
Stats
The paper does not provide any specific numerical data or statistics. The key results are the iteration complexity bounds and the comparison of the proposed algorithms with other methods on the joint diagonalization problem.
Quotes
None.

Deeper Inquiries

How can the proposed inexact Riemannian adaptive cubic regularization algorithm be extended to handle constrained optimization problems on Riemannian manifolds

The proposed inexact Riemannian adaptive cubic regularization algorithm can be extended to handle constrained optimization problems on Riemannian manifolds by incorporating constraints into the optimization process. One common approach is to utilize the concept of a projected gradient method, where at each iteration, the gradient is projected onto the tangent space of the manifold to ensure that the iterates remain feasible with respect to the constraints. This projection step enforces the constraints while still allowing for efficient optimization on the manifold. In the context of the algorithm proposed in the paper, the extension to handle constrained optimization would involve modifying the update steps to incorporate the projection onto the feasible set defined by the constraints. By incorporating the constraints into the optimization process, the algorithm can effectively navigate the constrained optimization landscape on the Riemannian manifold while still benefiting from the adaptive cubic regularization framework for optimization.

What are the potential limitations or drawbacks of the sub-sampling approach used to construct the inexact gradient and Hessian, and how can they be addressed

The sub-sampling approach used to construct the inexact gradient and Hessian in the algorithm may have some limitations or drawbacks that need to be addressed. One potential limitation is the trade-off between the accuracy of the approximations and the computational efficiency gained from sub-sampling. Sub-sampling can introduce errors in the estimation of the gradient and Hessian, which may impact the convergence properties of the algorithm. To address these limitations, several strategies can be employed: Adaptive Sampling: Implement adaptive sampling techniques that dynamically adjust the sample sizes based on the local geometry of the manifold or the convergence behavior of the optimization process. This can help balance the trade-off between accuracy and efficiency. Variance Reduction: Utilize variance reduction techniques to improve the quality of the approximations obtained through sub-sampling. Techniques such as control variates or importance sampling can help reduce the variance in the estimates and improve the overall accuracy. Regularization: Incorporate regularization techniques into the optimization process to mitigate the effects of noisy or inaccurate gradient and Hessian estimates. Regularization can help stabilize the optimization process and prevent divergence due to noisy estimates. By addressing these limitations and drawbacks, the sub-sampling approach can be enhanced to provide more reliable and accurate approximations of the gradient and Hessian, leading to improved convergence properties of the algorithm.

Can the ideas and techniques developed in this paper be applied to other types of optimization problems on Riemannian manifolds, such as those arising in machine learning, computer vision, or other scientific computing applications

The ideas and techniques developed in the paper can be applied to a wide range of optimization problems on Riemannian manifolds beyond the specific application of joint diagonalization discussed in the context. These techniques are particularly relevant in machine learning, computer vision, and scientific computing applications where optimization on manifolds is common. Machine Learning: In machine learning, optimization on Riemannian manifolds is prevalent in tasks such as dimensionality reduction, matrix factorization, and neural network training. The proposed algorithm can be adapted to handle optimization problems in these domains, improving convergence rates and efficiency. Computer Vision: Optimization on manifolds plays a crucial role in computer vision tasks such as image registration, shape analysis, and 3D reconstruction. By applying the inexact Riemannian adaptive cubic regularization algorithm, researchers can enhance optimization processes in computer vision applications, leading to more robust and accurate results. Scientific Computing: Optimization on manifolds is also essential in scientific computing applications such as signal processing, geodesy, and computational physics. The algorithm developed in the paper can be utilized to optimize functions on manifolds in these scientific domains, improving the efficiency and accuracy of the optimization process. By leveraging the concepts and methodologies presented in the paper, researchers and practitioners in these fields can enhance their optimization algorithms and address complex optimization challenges on Riemannian manifolds effectively.
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