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Accelerated Quadratic Proxy for Geometric Optimization

Basic Information

  • Title: Accelerated Quadratic Proxy for Geometric Optimization
  • Authors: Shahar Z. Kovalsky, Meirav Galun, Yaron Lipman
  • Affiliations: Weizmann Institute of Science
  • Publication Date: July 2016
  • Journal: ACM Transactions on Graphics, Volume 35, Number 4, Article 134
  • Keywords: Optimization, First-Order Methods, Acceleration, Preconditioning, Simplicial Meshes, Distortion, Geometry

Introduction

The paper introduces the Accelerated Quadratic Proxy (AQP), a first-order algorithm designed to optimize geometric energies defined over triangular and tetrahedral meshes. The optimization of geometric energies is common in computer graphics and geometric modeling but faces challenges such as slow convergence due to ill-conditioning. AQP addresses this by using a quadratic polynomial proxy whose Hessian is the Laplacian, effectively acting as a preconditioner and incorporating acceleration techniques.

Overview

AQP enhances convergence stability and rate by addressing the ill-conditioning prevalent in optimizing geometric energies over meshes. This approach demonstrates insensitivity to mesh resolution and converges in a nearly constant number of iterations across various test scenarios, a stark contrast to popular methods like Accelerated Gradient Descent and Quasi-Newton methods.

Summary

  • Problem Addressed: Slow convergence in the optimization of geometric energies over meshes due to ill-conditioning.
  • Proposed Solution: A first-order algorithm (AQP) that employs a quadratic polynomial proxy with the Laplacian as its Hessian for preconditioning and acceleration.
  • Results: AQP shows considerable speedup and reduced sensitivity to mesh resolution compared to traditional optimization methods in tasks like mesh deformation and surface parameterization.

Contributions

  • Introduction of a novel, efficient algorithm (AQP) for geometric optimization problems on meshes.
  • Demonstration of AQP's effectiveness in various geometric optimization tasks, showing considerable speedup over existing methods.
  • A theoretical and empirical analysis of the algorithm's performance, highlighting its advantages in terms of convergence speed and stability.

The paper situates AQP within the broader context of first-order optimization methods, comparing it against Newton methods, Quasi-Newton algorithms, Proximal methods, and other relevant techniques. It highlights AQP's novelty in leveraging the structure of geometric problems for improved optimization performance.

Limitations and Future Work

The authors note the need for more sophisticated line-search strategies and improvements in the implementation for further performance gains. Exploring the applicability of AQP to a broader range of energies and optimization problems in graphics and beyond is identified as an important direction for future research.

Details, Techniques, and Method

AQP iteratively approximates the solution to geometric optimization problems by combining acceleration techniques with a quadratic proxy minimization step, significantly improving the efficiency of each iteration. The method leverages the Laplacian for preconditioning, facilitating rapid convergence even for large-scale problems.

Describe all the Experiments

Experiments demonstrated AQP's superiority in mesh deformation and surface parameterization tasks compared to baseline methods like L-BFGS and AGD. AQP consistently required fewer iterations and less time to converge, showing its scalability and effectiveness across different problem sizes and types.

Conclusion

The Accelerated Quadratic Proxy method represents a significant advancement in the optimization of geometric energies over meshes, offering a scalable, efficient, and robust solution. Its success across various applications suggests wide applicability and potential for future exploration in geometric optimization and related fields.