mathematics

Ridge function

Ridge function 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 Ridge function rather than just read about it. In short: In mathematics, a ridge function is any function f : R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that can be written as the composition of an univariate function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } , that is called a profile function, with an affine transformation, given by a direction vector a ∈ R d {\displaystyle a\in \mathbb {R} ^{d}} with shift b ∈ R {\displays…

Key takeaways

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

Reference excerpt

In mathematics, a ridge function is any function f : R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that can be written as the composition of an univariate function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } , that is called a profile function, with an affine transformation, given by a direction vector a ∈ R d {\displaystyle a\in \mathbb {R} ^{d}} with shift b ∈ R {\displaystyle b\in \mathbb {R} } . Then, the ridge function reads

f ( x ) = g ( x ⊤ a + b ) {\displaystyle f(x)=g(x^{\top }a+b)} for x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} . Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp.

Relevance A ridge function is not susceptible to the curse of dimensionality, making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in d − 1 {\displaystyle d-1} directions: Let a 1 , … , a d − 1 {\displaystyle a_{1},\dots ,a_{d-1}} be d − 1 {\displaystyle d-1} independent vectors that are orthogonal to a {\displaystyle a} , such that these vectors span d − 1 {\displaystyle d-1} dimensions. Then

f ( x + ∑ k = 1 d − 1 c k a k ) = g ( x ⋅ a + ∑ k = 1 d − 1 c k a k ⋅ a ) = g ( x ⋅ a + ∑ k = 1 d − 1 c k 0 ) = g ( x ⋅ a ) = f ( x ) {\displaystyle f\left({\boldsymbol {x}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\cdot {\boldsymbol {a}}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}0\right)=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})=f({\boldsymbol {x}})}

for all c i ∈ R , 1 ≤ i < d {\displaystyle c_{i}\in \mathbb {R} ,1\leq i<d} . In other words, any shift of x {\displaystyle {\boldsymbol {x}}} in a direction perpendicular to a {\displaystyle {\boldsymbol {a}}} does not change the value of f {\displaystyle f} . Ridge functions play an essential role in amongst others projection pursuit, generalized linear models, and as activation functions in neural networks. For a survey on ridge functions, see. For books on ridge functions, see.

References

Worked examples

Example 1 — a first encounter with Ridge function

Start with the simplest possible case. Write down what Ridge function 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 Ridge function 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 Ridge function 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 Ridge function

In research: Ridge function 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 Ridge function 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: Ridge function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functions and mappings, so understanding it makes those chapters shorter.
In everyday life: Look for Ridge function 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 Ridge function in 20 minutes

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

Frequently asked questions

What is Ridge function in simple terms?

Why does Ridge function 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 Ridge function?

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 Ridge function.