Abstract
We show that the LSPIA method for curve and surface approximation, which was introduced by Deng and Lin (2014), 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.
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