Linear programming and l 1 regression: A geometric interpretation

Approaches to the solution of the l1 estimation problem have a long history, going back to at least Edgeworth in the nineteenth century. Modern solution approaches have been of two types: (i) descent methods and (ii) primal or dual LP methods. Simplex based algorithms have a standard geometric interpretation, but occasionally much more tangible geometric insights are available. For example, in the capacitated transhipment problem the workings of the LP solution algorithms can be interpreted on the underlying network. In this paper we develop geometric insight into the solution process directly in the space where the problem originates - the space of observations.