Between moving least-squares and moving least-ℓ1

Given function values at scattered points in R^d, possibly with noise, one of the ways of generating approximation to the function is by the method of moving least-squares (MLS). The method consists of computing local polynomials which approximate the data in a locally weighted least-squares sense. The resulting approximation is smooth, and is well approximating if the underlying function is smooth. Yet, as any least-squares based method it is quite sensitive to outliers in the data. It is well known that least-ℓ1 approximations are not sensitive to outliers. However, due to the nature of the ℓ1 norm, using it in the framework of a “moving” approximation will not give a smooth, or even a continuous approximation. This paper suggests an error measure which is between the ℓ1 and the ℓ2 norms, with the advantages of both. Namely, yielding smooth approximations which are not too sensitive to outliers. A fast iterative method for computing the approximation is demonstrated and analyzed. It is shown that for a scattered data taken from a smooth function, with few outliers, the new approximation gives an O(h) approximation error to the function.