LSPIA, (stochastic) gradient descent, and parameter correction. (English) Zbl 1482.65024
Summary: We show that the LSPIA method for curve and surface approximation, which was introduced by C. Deng and H. Lin, [“Progressive and iterative approximation for least squares B-spline curve and surface fitting”, Comput. Aided Des. 47, 32–44 (2014; doi:10.1016/j.cad.2013.08.012)], is equivalent to a gradient descent method. We also note that Deng and Lin’s results concerning feasible values of the stepsize are directly implied by classical results about convergence properties of the gradient descent method. We propose a modification based on stochastic gradient descent, which lends itself to a realization that employs the technology of neural networks. In addition, we show how to incorporate the optimization of the parameterization of the given data into this framework via parameter correction (PC). This leads to the new LSPIA-PC method and its neural-network based implementation. Numerical experiments indicate that it gives better results than LSPIA with comparable computational costs.
MSC:
| 65D10 | Numerical smoothing, curve fitting |
| 65D07 | Numerical computation using splines |
| 68T07 | Artificial neural networks and deep learning |
| 41A15 | Spline approximation |
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