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The method of **iteratively reweighted least squares** (**IRLS**) is used to solve certain optimization problems with objective functions of the form of a *p*-norm:

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 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 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.

IRLS can be used for ** ℓ_{1}** minimization and smoothed

To find the parameters * β* = (

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]}

- Feasible generalized least squares
- Weiszfeld's algorithm (for approximating the geometric median), which can be viewed as a special case of IRLS

**^**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. doi:10.1109/ICASSP.2008.4518498.**^**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:10.1007/978-0-387-70873-7. ISBN 978-0-387-70872-0.- ^
^{a}^{b}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

- Numerical Methods for Least Squares Problems by Åke Björck (Chapter 4: Generalized Least Squares Problems.)
- Practical Least-Squares for Computer Graphics. SIGGRAPH Course 11

- Solve under-determined linear systems iteratively