Minimum ℓ 1, ℓ 2, and ℓ ∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations

Abstract

Cadzow, J. A., Minimum ℓ 1, ℓ 2, and ℓ ∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations, Digital Signal Processing 12 (2002) 524–560 Many practical problems encountered in digital signal processing and other quantitative oriented disciplines entail finding a best approximate solution to an overdetermined system of linear equations. Invariably, the least squares error approximate solution (i.e., minimum ℓ 2 norm) is chosen for this task due primarily to the existence of a convenient closed expression for its determination. It should be noted, however, that in many applications a minimum ℓ 1 or ℓ ∞ norm approximate solution is preferable. For example, in cases where the data being analyzed contain a few data outliers a minimum ℓ 1 approximate solution is preferable since it tends to ignore bad data points. In other applications one may wish to determine an approximate solution whose largest error magnitude is the smallest possible (i.e., a minimum ℓ ∞ norm approximate solution). Unfortunately, there do not exist convenient closed form expressions for either the minimum ℓ 1 or the minimum ℓ ∞ norm approximate solution and one must resort to nonlinear programming methods for their determination. Effective algorithms for determining these two solutions are herein presented (see Cadzow, J. A., Data Analysis and Signal Processing: Theory and Applications).