Radial basis function kernel

Key takeaways

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

Reference excerpt

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two samples x , x ′ ∈ R k {\displaystyle \mathbf {x} ,\mathbf {x'} \in \mathbb {R} ^{k}} , represented as feature vectors in some input space, is defined as

K ( x , x ′ ) = exp ⁡ ( − ‖ x − x ′ ‖ 2 2 σ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)}

‖ x − x ′ ‖ 2 {\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}} may be recognized as the squared Euclidean distance between the two feature vectors. σ {\displaystyle \sigma } is a free parameter. An equivalent definition involves a parameter γ = 1 2 σ 2 {\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}} :

K ( x , x ′ ) = exp ⁡ ( − γ ‖ x − x ′ ‖ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})}

Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Radial basis function kernel

Start with the simplest possible case. Write down what Radial basis function kernel 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 Radial basis function kernel 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 Radial basis function kernel 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 Radial basis function kernel

In research: Radial basis function kernel 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 Radial basis function kernel 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: Radial basis function kernel is common in secondary-school and first-year university syllabi. It links to neighbouring topics Kernel methods for machine learning, Support vector machines, so understanding it makes those chapters shorter.
In everyday life: Look for Radial basis function kernel 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.

How to study Radial basis function kernel in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Radial basis function kernel 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 Radial basis function kernel out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Radial basis function kernel in simple terms?

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.

Why does Radial basis function kernel 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 Radial basis function kernel?

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 Radial basis function kernel.