science

Iteratively reweighted least squares

Iteratively reweighted least squares is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Iteratively reweighted least squares rather than just read about it. In short: 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, a r g m i n β ⁡ ∑ i = 1 n | y i − f i ( β ) | p , {\displaystyle \operatorname {arg\,min} _{\boldsymbol {\beta }}\sum _{i=1}^{n}{\big |}y_{i}-f_{i}({\boldsymbol {\beta }}){\big |}^{p},} by an iterative method in which each step involves solving a weighted least sq…

Key takeaways

Iteratively reweighted least squares belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Iteratively reweighted least squares to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Iteratively reweighted least squares from memory before moving on to harder problems.

Reference excerpt

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,

a r g m i n β ⁡ ∑ i = 1 n | y i − f i ( β ) | p , {\displaystyle \operatorname {arg\,min} _{\boldsymbol {\beta }}\sum _{i=1}^{n}{\big |}y_{i}-f_{i}({\boldsymbol {\beta }}){\big |}^{p},}

by an iterative method in which each step involves solving a weighted least squares problem of the form

β ( t + 1 ) = a r g m i n β ⁡ ∑ i = 1 n w i ( β ( t ) ) | y i − f i ( β ) | 2 . {\displaystyle {\boldsymbol {\beta }}^{(t+1)}=\operatorname {arg\,min} _{\boldsymbol {\beta }}\sum _{i=1}^{n}w_{i}({\boldsymbol {\beta }}^{(t)}){\big |}y_{i}-f_{i}({\boldsymbol {\beta }}){\big |}^{2}.}

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.

Examples

L1 minimization for sparse recovery IRLS can be used for ℓ1 minimization and smoothed ℓp minimization, p < 1, in 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.

Lp norm linear regression To find the parameters β = (β1, …,βk)T which minimize the Lp norm for the linear regression problem,

a r g m i n β ‖ y − X β ‖ p = a r g m i n β ∑ i = 1 n | y i − X i β | p , {\displaystyle {\underset {\boldsymbol {\beta }}{\operatorname {arg\,min} }}{\big \|}\mathbf {y} -X{\boldsymbol {\beta }}\|_{p}={\underset {\boldsymbol {\beta }}{\operatorname {arg\,min} }}\sum _{i=1}^{n}\left|y_{i}-X_{i}{\boldsymbol {\beta }}\right|^{p},}

the IRLS algorithm at step t + 1 involves solving the weighted linear least squares problem

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Iteratively reweighted least squares

Start with the simplest possible case. Write down what Iteratively reweighted least squares claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Iteratively reweighted least squares before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Iteratively reweighted least squares ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Iteratively reweighted least squares

In research: Iteratively reweighted least squares appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Iteratively reweighted least squares in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Iteratively reweighted least squares is common in secondary-school and first-year university syllabi. It links to neighbouring topics Least squares, so understanding it makes those chapters shorter.
In everyday life: Look for Iteratively reweighted least squares outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Iteratively reweighted least squares in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Iteratively reweighted least squares means in your own words.
Compare your version with the excerpt and mark what you missed.
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Frequently asked questions

What is Iteratively reweighted least squares in simple terms?

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, a r g m i n β ⁡ ∑ i = 1 n | y i − f i ( β ) | p , {\displaystyle \operatorname {arg\,min} _{\boldsymbol {\beta }}\sum _{i=1}^{n}{\big |}y_{i}-f_…

Why does Iteratively reweighted least squares matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Iteratively reweighted least squares?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Iteratively reweighted least squares.