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 of a pnorm:
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by an iterative method in which each step involves solving a weighted least squares problem of the form:[1]
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an Mestimator, as a way of mitigating the influence of outliers in an otherwise normallydistributed data set. For example, by minimizing the least absolute errors rather than the least square errors.
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.
Examples
L_{1} minimization for sparse recovery
IRLS can be used for ℓ_{1} minimization and smoothed ℓ_{p} minimization, p < 1, in compressed sensing problems. It has been proved that the algorithm has a linear rate of convergence for ℓ_{1} norm and superlinear for ℓ_{t} with t < 1, under the restricted isometry property, which is generally a sufficient condition for sparse solutions.[2][3] However, in most practical situations, the restricted isometry property is not satisfied.
L^{p} norm linear regression
To find the parameters β = (β_{1}, …,β_{k})^{T} which minimize the L^{p} norm for the linear regression problem,
the IRLS algorithm at step t + 1 involves solving the weighted linear least squares problem:[4]
where W^{(t)} is the diagonal matrix of weights, usually with all elements set initially to:
and updated after each iteration to:
In the case p = 1, this corresponds to least absolute deviation regression (in this case, the problem would be better approached by use of linear programming methods,[5] so the result would be exact) and the formula is:
To avoid dividing by zero, regularization must be done, so in practice the formula is:
where is some small value, like 0.0001.[5] Note the use of in the weighting function is equivalent to the Huber loss function in robust estimation. [6]
See also
 Feasible generalized least squares
 Weiszfeld's algorithm (for approximating the geometric median), which can be viewed as a special case of IRLS
Notes
 C. Sidney Burrus, Iterative Reweighted Least Squares
 Chartrand, R.; Yin, W. (March 31 – April 4, 2008). "Iteratively reweighted algorithms for compressive sensing". IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2008. pp. 3869–3872.
 Daubechies, I.; Devore, R.; Fornasier, M.; Güntürk, C. S. N. (2010). "Iteratively reweighted least squares minimization for sparse recovery". Communications on Pure and Applied Mathematics. 63: 1–38. arXiv:0807.0575. doi:10.1002/cpa.20303.
 Gentle, James (2007). "6.8.1 Solutions that Minimize Other Norms of the Residuals". Matrix algebra. Springer Texts in Statistics. New York: Springer. doi:10.1007/9780387708737. ISBN 9780387708720.
 William A. Pfeil, Statistical Teaching Aids, Bachelor of Science thesis, Worcester Polytechnic Institute, 2006
 Fox, J.; Weisberg, S. (2013),Robust Regression, Course Notes, University of Minnesota
References
 Numerical Methods for Least Squares Problems by Åke Björck (Chapter 4: Generalized Least Squares Problems.)
 Practical LeastSquares for Computer Graphics. SIGGRAPH Course 11