000205806 001__ 205806
000205806 005__ 20190317000123.0
000205806 037__ $$aREP_WORK
000205806 245__ $$aA New Identity for the Least-square Solution of Overdetermined Set of Linear Equations
000205806 269__ $$a2015
000205806 260__ $$c2015
000205806 336__ $$aReports
000205806 520__ $$aIn 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.
000205806 6531_ $$aOver-determined linear equation
000205806 6531_ $$aLeast square solution
000205806 700__ $$aHaghighatshoar, Saeid
000205806 700__ $$0246036$$g193629$$aTaghizadeh, Mohammadjavad
000205806 700__ $$aAsaei, Afsaneh$$g188259$$0243353
000205806 8564_ $$uhttps://infoscience.epfl.ch/record/205806/files/overdetermined_equations.pdf$$zn/a$$s200648$$yn/a
000205806 909C0 $$xU10381$$0252189$$pLIDIAP
000205806 909CO $$ooai:infoscience.tind.io:205806$$qGLOBAL_SET$$pSTI$$preport
000205806 917Z8 $$x188259
000205806 937__ $$aEPFL-REPORT-205806
000205806 973__ $$aEPFL
000205806 980__ $$aREPORT