Sparse and nonnegative sparse D-MORPH regression

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.