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

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 l 1 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 in [35], 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.

Piecewise Sparse Recovery in Unions of Bases

“…The coefficient vector x = (x T n , x T t ) T is "piecewise" sparse vector. Another example is the problem of reconstructing a surface from scattered data in approximation space H = N i=1 H j , where H j ⊆ H j+1 are principal shift invariant (PSI) spaces generated by a single compactly supported function [29], the fitting surface is g…”

“…is the vector to be determined. Due to the property of PSI spaces, the coefficients to be determined by l 1 minimization in DRAFT [29] are "piecewise" sparse structured, i.e. each c i ∈ R n i is a sparse vector in H i .…”

Distributed proximal gradient algorithm for non-smooth non-convex optimization over time-varying networks

“…In this problem setup, each node in the network only knows local function information and communicates with its neighbors to solve the global optimization problem. One fundamental model for distributed non-smooth non-convex optimization, arising from optimization problems such as Lasso [1] , SVM [2] , and optimizing neural networks [3] , is that each local objective function of a node is the summation of a (non-convex) differentiable function and a non-smooth convex function (l 1 norm or indicator function). Although the research on distributed optimization has made significant progress on non-smooth convex problems [4]- [8] , distributed non-smooth non-convex optimization is still challenging.…”