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 [BibTeX] [Marc21]
A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations
Type of publication: Idiap-RR
Citation: Haghighatshoar_Idiap-RR-35-2015
Number: Idiap-RR-35-2015
Year: 2015
Month: 12
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.
Keywords: Least square solution., Over-determined linear equation
Projects Idiap
FP 7
Authors Haghighatshoar, Saeid
Taghizadeh, Mohammad J.
Asaei, Afsaneh
Added by: [ADM]
Total mark: 0
  • Haghighatshoar_Idiap-RR-35-2015.pdf (MD5: 747169292b30d5a4da5079e0a0c033d3)