Best fitting straight line

Consider the straight line that best fits a collection of data points. Suppose there are four points. Two of them are (-10,-10), (-10,10). I'll call these two data points the ``left slot''. The other two are (10,-10), (10,10) which I'll call the ``right slot''. You can easily see that straight lines that lie within both the left slot and the right slot all have the same sum of absolute distances from the line to the data. For each slot the sum is 20 so the total is 40. Thus L₁ gives us many lines inside the slots but it does not select any particular line. (This example is said to come from Albert Tarantola.)

You might object to having two data points at the same coordinate. By moving them apart a little, we suppose the ``degeneracy'' is broken, that a unique line becomes defined. Perhaps so. Never the less, it is clear that the residual is ``almost minimum'' for all lines inside the slots, and it is much bigger for lines outside the slots. Thus the reality of the slots remains, even where technically we might avoid them. Again, L₁ has the appealing feature that an additional data point, even if weighted weakly, could break the ambiguity.