Growth function

Key takeaways

Growth 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 Growth function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Growth function from memory before moving on to harder problems.

Reference excerpt

The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods. The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties. It is a basic concept in machine learning.

Definitions

Set-family definition Let H {\displaystyle H} be a set family (a set of sets) and C {\displaystyle C} a set. Their intersection is defined as the following set-family:

H ∩ C := { h ∩ C ∣ h ∈ H } {\displaystyle H\cap C:=\{h\cap C\mid h\in H\}}

The intersection-size (also called the index) of H {\displaystyle H} with respect to C {\displaystyle C} is | H ∩ C | {\displaystyle |H\cap C|} . If a set C m {\displaystyle C_{m}} has m {\displaystyle m} elements then the index is at most 2 m {\displaystyle 2^{m}} . If the index is exactly 2m then the set C {\displaystyle C} is said to be shattered by H {\displaystyle H} , because H ∩ C {\displaystyle H\cap C} contains all the subsets of C {\displaystyle C} , i.e.:

| H ∩ C | = 2 | C | , {\displaystyle |H\cap C|=2^{|C|},}

The growth function measures the size of H ∩ C {\displaystyle H\cap C} as a function of | C | {\displaystyle |C|} . Formally:

Growth ⁡ ( H , m ) := max C : | C | = m | H ∩ C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H\cap C|}

Hypothesis-class definition Equivalently, let H {\displaystyle H} be a hypothesis-class (a set of binary functions) and C {\displaystyle C} a set with m {\displaystyle m} elements. The restriction of H {\displaystyle H} to C {\displaystyle C} is the set of binary functions on C {\displaystyle C} that can be derived from H {\displaystyle H} :

H C := { ( h ( x 1 ) , … , h ( x m ) ) ∣ h ∈ H , x i ∈ C } {\displaystyle H_{C}:=\{(h(x_{1}),\ldots ,h(x_{m}))\mid h\in H,x_{i}\in C\}}

The growth function measures the size of H C {\displaystyle H_{C}} as a function of | C | {\displaystyle |C|} :

Growth ⁡ ( H , m ) := max C : | C | = m | H C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H_{C}|}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Growth function

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

In research: Growth 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 Growth 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: Growth function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational learning theory, Families of sets, Measures of complexity, so understanding it makes those chapters shorter.
In everyday life: Look for Growth 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.

How to study Growth function in 20 minutes

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

Frequently asked questions

What is Growth function in simple terms?

The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods.

Why does Growth 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 Growth 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 Growth function.