A class of multilevel structured low-rank approximation arising in material processing

ABSTRACT

This work is devoted to studying the problem of approximating a given matrix by the product of two low-rank structured matrix which arises in the material processing. Firstly, we analyse some properties of the original problem and utilize the alternating least squares method to reformulate it into two subproblems. Then by using the Gramian representation and a Vandermonde-like transformation, the feasible sets of the subproblems are characterized, which makes them be respectively transformed into two unconstrained minimization problems. Finally, we derive the expressions of the gradients of the objective functions and apply the gradient descent algorithm to solve them. Numerical examples are tested to show that our method performs well in terms of the residual error of the objective function.

Keywords:

• Low-rank constraint
• structured matrix
• Gramian representation
• Vandermonde-like transformation
• material processing

2010 AMS subject classifications:

• 11D07
• 15B99
• 41A29
• 68W25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [Nos. 11671318; 11171270; 51471126], the Natural Science Foundation of Shaanxi Province [No. 2014JM1029], the Fundamental Research Funds for the Central Universities [Xi'an Jiaotong University, No. xkjc2014008], and China Postdoctoral Science Foundation [No. 2014M552435].