mathematics

Representer theorem

Representer theorem is a mathematics 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 Representer theorem rather than just read about it. In short: For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{*}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. Formal statement The following Representer…

Key takeaways

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

Reference excerpt

For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{*}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.

Formal statement The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } on a non-empty set X {\displaystyle {\mathcal {X}}} with a corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} . Let there be given

a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{n}\to \mathbb {R} \cup \lbrace \infty \rbrace } , which together define the following regularized empirical risk functional on H k {\displaystyle H_{k}} :

f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . {\displaystyle f\mapsto E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right).}

Then, any minimizer of the empirical risk

f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) } , ( ∗ ) {\displaystyle f^{*}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad (*)}

admits a representation of the form:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Representer theorem

Start with the simplest possible case. Write down what Representer theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Representer theorem 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 Representer theorem 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 Representer theorem

In research: Representer theorem appears in mathematics 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 Representer theorem 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: Representer theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational learning theory, Hilbert spaces, Theoretical computer science, so understanding it makes those chapters shorter.
In everyday life: Look for Representer theorem 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Representer theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Representer theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Representer theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Representer theorem in simple terms?

Why does Representer theorem matter?

Because it connects several mathematics 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 Representer theorem?

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 Representer theorem.