Hyper basis function network

Key takeaways

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

Reference excerpt

In machine learning, a Hyper basis function network, or HyperBF network, is a generalization of radial basis function (RBF) networks concept, where the Mahalanobis-like distance is used instead of the Euclidean distance measure. Hyper basis function networks were first introduced by Poggio and Girosi in the 1990 paper “Networks for Approximation and Learning”.

Network Architecture The typical HyperBF network structure consists of a real input vector x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , a hidden layer of activation functions and a linear output layer. The output of the network is a scalar function of the input vector, ϕ : R n → R {\displaystyle \phi :\mathbb {R} ^{n}\to \mathbb {R} } , is given by

where N {\displaystyle N} is a number of neurons in the hidden layer, μ j {\displaystyle \mu _{j}} and a j {\displaystyle a_{j}} are the center and weight of neuron j {\displaystyle j} . The activation function ρ j ( | | x − μ j | | ) {\displaystyle \rho _{j}(||x-\mu _{j}||)} at the HyperBF network takes the following form

where R j {\displaystyle R_{j}} is a positive definite d × d {\displaystyle d\times d} matrix. Depending on the application, the following types of matrices R j {\displaystyle R_{j}} are usually considered

R j = 1 2 σ 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma ^{2}}}\mathbb {I} _{d\times d}} , where σ > 0 {\displaystyle \sigma >0} . This case corresponds to the regular RBF network.

R j = 1 2 σ j 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma _{j}^{2}}}\mathbb {I} _{d\times d}} , where σ j > 0 {\displaystyle \sigma _{j}>0} . In this case, the basis functions are radially symmetric, but are scaled with different width.

R j = d i a g ( 1 2 σ j 1 2 , . . . , 1 2 σ j z 2 ) I d × d {\displaystyle R_{j}=diag\left({\frac {1}{2\sigma _{j1}^{2}}},...,{\frac {1}{2\sigma _{jz}^{2}}}\right)\mathbb {I} _{d\times d}} , where σ j i > 0 {\displaystyle \sigma _{ji}>0} . Every neuron has an elliptic shape with a varying size. Positive definite matrix, but not diagonal.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hyper basis function network

Start with the simplest possible case. Write down what Hyper basis function network claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer 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 Hyper basis function network 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 Hyper basis function network 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 Hyper basis function network

In research: Hyper basis function network appears in computer 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 Hyper basis function network 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: Hyper basis function network is common in secondary-school and first-year university syllabi. It links to neighbouring topics Artificial neural networks, Classification algorithms, Machine learning algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Hyper basis function network 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 Hyper basis function network in 20 minutes

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

Frequently asked questions

What is Hyper basis function network in simple terms?

In machine learning, a Hyper basis function network, or HyperBF network, is a generalization of radial basis function (RBF) networks concept, where the Mahalanobis-like distance is used instead of the Euclidean distance measure. Hyper basis function networks were first introduced by Poggio and Giro…

Why does Hyper basis function network matter?

Because it connects several computer 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 Hyper basis function network?

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 Hyper basis function network.