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In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation , where is an full-rank matrix, is a column-vector of dimension , and (the number of equations) is larger than or equal to (the dimension of the unknown vector ). Generally, the equations are inconsistent and there is no feasible solution for unless belongs to the column-span of . In the least-square approach, a candidate solution is found as the unique that minimizes the error function . We propose a more general approach that consist in considering all the consistent subset of the equations, finding their solutions, and taking a weighted average of them to build a candidate solution. In particular, we show that by weighting the solutions with the squared determinant of their coefficient matrix, the resulting candidate solution coincides with the least square solution.
Rachid Guerraoui, Nirupam Gupta, Youssef Allouah, Geovani Rizk, Rafaël Benjamin Pinot
Alexandre Caboussat, Dimitrios Gourzoulidis