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The connections between the error function used in multilinear regression and the expected, or assumed, properties of the data are investigated. It is shown that two of the most basic properties often required in data analysis, scale and rotational invariance, are incompatible. With this, it is established that multilinear regression using an error function derived from a geometric mean is both scale and reflectively invariant. The resulting error function is also shown to have the property that its minimizer, under certain conditions, is well approximated using the centroid of the error simplex. It is then applied to several multidimensional real world data sets, and compared to other regression methods.