Sparse and nonnegative sparse D-MORPH regression

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• 作者：Genyuan Li ; Roberto Rey-de-Castro ; Xi Xing…
• 关键词：Underdetermined system ; D ; MORPH regression ; Least ; squares regression ; IRLS ; RRLS ; Quantum ; control ; mechanism identification ; Mass spectrum analysis
• 刊名：Journal of Mathematical Chemistry
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：53
• 期：8
• 页码：1885-1914
• 全文大小：2,937 KB
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• 作者单位：Genyuan Li (1)
  Roberto Rey-de-Castro (1)
  Xi Xing (1)
  Herschel Rabitz (1)
  
  1. Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA
  
• 刊物类别：Chemistry and Materials Science
• 刊物主题：Chemistry
  Physical Chemistry
  Theoretical and Computational Chemistry
  Mathematical Applications in Chemistry
  
• 出版者：Springer Netherlands
• ISSN：1572-8897

文摘

An underdetermined linear algebraic equation system \(\mathbf{y}={\varvec{\Phi }}\mathbf{x}\), where \({\varvec{\Phi }}\) is an \(m\times n (m<n)\) rectangular constant matrix with rank \(r\le m\) and \(\mathbf{y}\in \mathrm {Ran}({\varvec{\Phi }})\) (range of \({\varvec{\Phi }})\), has an infinite number of solutions. Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression seeks a solution satisfying the extra requirement of minimizing a chosen cost function, \({\mathcal {K}}\). A wide variety of choices of the cost function makes it possible to achieve diverse goals, and hence D-MORPH regression has been successfully applied to solve a range of problems. In this paper, D-MORPH regression is extended to determine a sparse or a nonnegative sparse solution of the vector \(\mathbf{x}\). For this purpose, recursive reweighted least-squares (RRLS) minimization is adopted and modified to construct the cost function \({\mathcal {K}}\) for D-MORPH regression. The advantage of sparse and nonnegative sparse D-MORPH regression is that the matrix \({\varvec{\Phi }}\) does not need to have row-full rank, thereby enabling flexibility to search for sparse solutions \(\mathbf{x}\) with ancillary properties in practical applications. These tools are applied to (a) simulation data for quantum-control-mechanism identification utilizing high dimensional model representation (HDMR) modeling and (b) experimental mass spectral data for determining the composition of an unknown mixture of chemical species.