# Sparse approximate solution of fitting surface to scattered points by MLASSO model

## TL;DR

This paper presents a multilevel LASSO (MLASSO) model for efficiently obtaining sparse surface approximations to scattered data, balancing data fit and sparsity, and highlighting high-gradient regions.

## Contribution

The paper introduces a novel MLASSO model solved by ADMM for sparse surface fitting, improving upon existing methods in efficiency and accuracy.

## Key findings

- MLASSO provides a good trade-off between data mismatch and sparsity.
- The model effectively captures high-gradient surface regions.
- Numerical experiments demonstrate superior performance over AGLASSO and MBA algorithms.

## Abstract

The goal of this paper is to achieve a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant (PSI) space and the l1 norm minimization. In order to obtain different sparsity of the approximation solution, the problem is represented as a multilevel LASSO (MLASSO) model with different regularization parameters. The MLASSO model can be solved efficiently by the alternating direction method of multipliers. Numerical experiments indicate that compared to the AGLASSO model and the basic MBA algorithm, the MLASSO model can provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution. Moreover, the solution by the MLASSO model can reflect the regions of the underlying surface where high gradients occur.