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@TECHREPORT{Haghighatshoar_Idiap-RR-35-2015,
         author = {Haghighatshoar, Saeid and Taghizadeh, Mohammad J. and Asaei, Afsaneh},
       keywords = {Least square solution., Over-determined linear equation},
       projects = {Idiap, FP 7},
          month = {12},
          title = {A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations},
           type = {Idiap-RR},
         number = {Idiap-RR-35-2015},
           year = {2015},
    institution = {Idiap},
       abstract = {In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation $Ax=b$, where $A$ is an $m\times n$ full-rank matrix, $b$ is a column-vector of dimension $m$, and $m$ (the number of equations) is larger than or equal to  $n$ (the dimension of the unknown vector $x$). Generally, the equations are inconsistent and there is no feasible solution for $x$ unless $b$ belongs to the column-span of $A$. In the least-square approach, a candidate solution is found as the unique $x$ that minimizes the error function $\|Ax-b\|_2$. 

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.},
            pdf = {https://publications.idiap.ch/attachments/reports/2015/Haghighatshoar_Idiap-RR-35-2015.pdf}
}