The objective is to accurately grasp the overall picture of the target object from the observed data. However, the observed data is sparse and buried in noise. This challenge has long been known as an "inverse problem", but the accurate estimation of true value is not always easy. In recent years, a least squares estimation method incorporating regularization, known as sparse modeling, has been attracting attention. It has been successful in estimation where the sparsity of vectors and the low rank of matrices can be assumed.

In terms of regularization terms, discrete value functions such as the L0 pseudo-norm and matrix rank can be considered. However, since the optimization problem is NP-hard, the best approximate convex functions of discrete value functions, such as the L1 norm and nuclear norm, are substituted. The well-known LASSO is an example of such sparse modeling methods. Subsequently, a non-convex regularization function (LiGME regularization function) that parametrically connects the discrete value function and the best approximate convex function has been proposed. The overall convexity condition of this model was clarified, and a global optimization algorithm was realized.

The LiGME method can approach non-convex regularization, which is an ideal indicator of sparsity, while maintaining the convexity of the entire least squares estimation model. This allows for the exploration of the global optimal solution of the model. In this process, how to properly design the Generalized Moreau Enhancement (GME) matrix is key to ensuring overall convexity. Until now, the design of the GME matrix has been limited to special cases where the sparsity manifestation matrix is full row rank, thus narrowing the application range of the LiGME method.

In contrast, this paper realizes an algebraic method that can design a GME matrix in a finite number of steps (the same computational complexity as LU decomposition) for any sparsity manifestation matrix. This allows for the design of a GME matrix that guarantees the overall convexity of the objective function with overwhelmingly less computational complexity than conventional methods, without the need for eigenvalue decomposition or iterative calculations. Based on the method of this paper, numerous applications that are beneficial in engineering are expected for various problems where sparsity or group sparsity can be assumed.