Iteratively reweighted least squares
The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form:
by an iterative method in which each step involves solving a weighted least squares problem of the form:
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set. For example, by minimizing the least absolute error rather than the least square error.
Although not a linear regression problem, Weiszfeld's algorithm for approximating the geometric median can also be viewed as a special case of iteratively reweighted least squares, in which the objective function is the sum of distances of the estimator from the samples.
One of the advantages of IRLS over linear programming and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms.
IRLS can be used for ℓ1 minimization and smoothed ℓp minimization, p < 1, in the compressed sensing problems. It has been proved that the algorithm has a linear rate of convergence for ℓ1 norm and superlinear for ℓt with t < 1, under the restricted isometry property, which is generally a sufficient condition for sparse solutions. However in most practical situations, the restricted isometry property is not satisfied.
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