Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds

Core Concepts

The author presents a modified memoryless quasi-Newton method based on the spectral-scaling Broyden family on Riemannian manifolds, focusing on global convergence and Wolfe conditions.

Abstract

The paper introduces a novel approach to optimization problems on Riemannian manifolds using memoryless quasi-Newton methods. It compares various methods, including the proposed algorithm, with numerical experiments showcasing superior performance under different parameter settings.

The study delves into the theoretical foundations of Riemannian optimization and proposes an innovative algorithm based on the spectral-scaling Broyden family. By leveraging concepts like retraction and vector transport, the method aims to optimize off-diagonal cost functions efficiently.

Key highlights include the formulation of search directions satisfying sufficient descent conditions, global convergence analyses under specific conditions, and comparisons with existing algorithms. The research emphasizes practical applications in solving complex optimization problems on manifolds.

Overall, the paper contributes valuable insights into memoryless quasi-Newton methods tailored for Riemannian optimization challenges. It underscores the significance of parameter selection and algorithm design in achieving optimal convergence rates and computational efficiency.

Stats

γk−1 > 0 is a sizing parameter.
τk−1 > 0 is a spectral-scaling parameter.
ξk−1 ∈ [0, 1] is a parameter.
ν∥sk−1∥2xk ≤ s♭ k−1zk−1.
∥zk−1∥xk ≤ ν∥sk−1∥xk.

Quotes

"The idea behind the memoryless spectral-scaling Broyden family is simple: replace Hk−1 with idTxk M."
"Algorithm 1 converges in the sense that lim inf ∥gk∥xk = 0 holds."
"The study delves into the theoretical foundations of Riemannian optimization and proposes an innovative algorithm based on the spectral-scaling Broyden family."

Key Insights Distilled From

Modified memoryless spectral-scaling Broyden family on Riemannian manifolds

by Hiroyuki Sak... at arxiv.org 03-11-2024

https://arxiv.org/pdf/2307.08986.pdf

Deeper Inquiries

How can memoryless quasi-Newton methods be further optimized for specific types of optimization problems

Memoryless quasi-Newton methods can be further optimized for specific types of optimization problems by fine-tuning the parameters and modifications used in the algorithm. For example, adjusting the spectral-scaling parameter τk-1 and the sizing parameter γk-1 can have a significant impact on convergence speed and solution accuracy. Additionally, exploring different choices for the search direction update formula, such as incorporating adaptive strategies based on problem characteristics or historical information, can enhance the performance of memoryless quasi-Newton methods for specific optimization tasks.

What are potential limitations or drawbacks of using memoryless spectral-scaling Broyden families in practical applications

One potential limitation of using memoryless spectral-scaling Broyden families in practical applications is their sensitivity to parameter settings. The choice of parameters such as φk-1, ξk-1, and zk-1 can significantly affect algorithm performance, requiring careful tuning for optimal results. Moreover, these methods may require more computational resources compared to simpler optimization techniques due to their iterative nature and additional calculations involved in updating search directions with spectral scaling.

How does this research contribute to advancements in machine learning algorithms beyond traditional optimization techniques

This research contributes to advancements in machine learning algorithms beyond traditional optimization techniques by introducing modified memoryless quasi-Newton methods based on spectral-scaling Broyden families on Riemannian manifolds. By leveraging concepts from Riemannian geometry and optimization theory, this approach offers a novel perspective on solving unconstrained optimization problems with global convergence guarantees under certain conditions. These advanced algorithms provide efficient solutions for complex optimization tasks commonly encountered in machine learning applications like low-rank tensor completion, shape analysis, and other high-dimensional data processing scenarios.