Integrating adaptive optimization into least squares progressive iterative approximation

TL;DR

The paper addresses the efficiency of surface fitting with LSPIA by introducing AdagradLSPIA, which injects Adagrad-style adaptive per-coordinate learning rates into the LSPIA framework. By updating control-point adjustments with adaptive weights derived from accumulated gradients, the method accelerates convergence while preserving convergence guarantees for convex least-squares problems. Experimental results on tensor-product B-spline surface fitting (including noise) show that AdagradLSPIA achieves higher accuracy, faster convergence, and robustness to global weight choices, with surface quality comparable to or better than LSPIA. The approach maintains linear per-iteration complexity and offers practical benefits for geometric modeling tasks in CAD and reverse engineering.

Abstract

This paper introduces the Adaptive Gradient Least Squares Progressive iterative Approximation (AdagradLSPIA), an accelerated version of the Least Squares Progressive Iterative Approximation (LSPIA) method, enhanced with adaptive optimization techniques inspired by the adaptive gradient (Adagrad) algorithm. By using historical (accumulated) gradient information to dynamically adjust weights, AdagradLSPIA achieves faster convergence compared to the standard LSPIA method. The effectiveness of AdagradLSPIA is demonstrated through its application to tensor product B-spline surface fitting, where this method consistently outperforms LSPIA in terms of accuracy, computational efficiency, and robustness to variations in global weight selection.

Figures (6)

• Figure 1: Data points representing an automobile hood model, including noise-free data (left) and noisy data (right) with added zero-mean Gaussian noise (standard deviation 0.02).
• Figure 2: Comparison of the LSPIA and AdagradLSPIA methods for noise-free data across a range of global weight values. The plots depict the variations in fitting error, elapsed time (in s), and the number of iterations required to converge, highlighting the broader convergence region and superior performance of AdagradLSPIA.
• Figure 3: Comparison of the LSPIA and AdagradLSPIA methods for noisy data across a range of global weight values. The plots depict the variations in fitting error, elapsed time (in s), and the number of iterations required to converge, highlighting the broader convergence region and superior performance of AdagradLSPIA.
• Figure 4: Convergence plots for the LSPIA and AdagradLSPIA methods using optimized global weight values. The plots show the reduction in fitting error over elapsed time and iterations for both noise-free and noisy data, demonstrating the faster convergence of AdagradLSPIA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
• Figure 5: The final fitting surfaces generated by the LSPIA and AdagradLSPIA methods for noise-free and noisy data. The surfaces were obtained using global weights that minimize the fitting error (\ref{['tab:1']}).