A New Identity for the Leastsquare Solution of Overdetermined Set of Linear Equations
Type of publication:  IdiapRR 
Citation:  Haghighatshoar_IdiapRR352015 
Number:  IdiapRR352015 
Year:  2015 
Month:  12 
Institution:  Idiap 
Abstract:  In this paper, we prove a new identity for the leastsquare solution of an overdetermined set of linear equation $Ax=b$, where $A$ is an $m\times n$ fullrank matrix, $b$ is a columnvector 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 columnspan of $A$. In the leastsquare approach, a candidate solution is found as the unique $x$ that minimizes the error function $\Axb\_2$. 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. 
Keywords:  Least square solution., Overdetermined linear equation 
Projects 
Idiap FP 7 
Authors  
Added by:  [ADM] 
Total mark:  0 
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