# Moving least squares

Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested.

In computer graphics, the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling.

In numerical analysis to handle contributions of geometry where it is difficult to obtain discretizations, the moving least squares methods have also been used and generalized to solve PDEs on curved surfaces and other geometries.[1][2][3] This includes numerical methods developed for curved surfaces for solving scalar parabolic PDEs [1] [3] and vector-valued hydrodynamic PDEs.[2]

In machine learning, moving least squares methods have also been used to develop model classes and learning methods. This includes function regression methods [4] and neural network function and operator regression approaches, such as GMLS-Nets.[5]

## Definition

Here is a 2D example. The circles are the samples and the polygon is a linear interpolation. The blue curve is a smooth approximation of order 3.

Consider a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ and a set of sample points ${\displaystyle S=\{(x_{i},f_{i})|f(x_{i})=f_{i}\}}$. Then, the moving least square approximation of degree ${\displaystyle m}$ at the point ${\displaystyle x}$ is ${\displaystyle {\tilde {p}}(x)}$ where ${\displaystyle {\tilde {p}}}$ minimizes the weighted least-square error

${\displaystyle \sum _{i\in I}(p(x_{i})-f_{i})^{2}\theta (\|x-x_{i}\|)}$

over all polynomials ${\displaystyle p}$ of degree ${\displaystyle m}$ in ${\displaystyle \mathbb {R} ^{n}}$. ${\displaystyle \theta (s)}$ is the weight and it tends to zero as ${\displaystyle s\to \infty }$.

In the example ${\displaystyle \theta (s)=e^{-s^{2}}}$. The smooth interpolator of "order 3" is a quadratic interpolator.

## References

1. ^ a b Liang, Jian; Zhao, Hongkai (January 2013). "Solving Partial Differential Equations on Point Clouds". SIAM Journal on Scientific Computing. 35 (3): A1461–A1486. doi:10.1137/120869730.
2. ^ a b Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J. (15 May 2020). "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach". Journal of Computational Physics. 409: 109340. doi:10.1016/j.jcp.2020.109340.
3. ^ a b Gross, B. J.; Kuberry, P.; Atzberger, P. J. (15 March 2022). "First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)". Journal of Computational Physics. 453: 110932. doi:10.1016/j.jcp.2021.110932. ISSN 0021-9991.
4. ^ Wang, Hong-Yan; Xiang, Dao-Hong; Zhou, Ding-Xuan (1 March 2010). "Moving least-square method in learning theory". Journal of Approximation Theory. 162 (3): 599–614. doi:10.1016/j.jat.2009.12.002. ISSN 0021-9045.
5. ^ Trask, Nathaniel; Patel, Ravi G.; Gross, Ben J.; Atzberger, Paul J. (13 September 2019). "GMLS-Nets: A framework for learning from unstructured data". arXiv:1909.05371 [physics, stat].