Random feature

Key takeaways

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

Reference excerpt

Random features (RF) are a technique used in machine learning to approximate kernel methods, introduced by Ali Rahimi and Ben Recht in their 2007 paper "Random Features for Large-Scale Kernel Machines", and extended by. RF uses a Monte Carlo approximation to kernel functions by randomly sampled feature maps. It is used for datasets that are too large for traditional kernel methods like support vector machine, kernel ridge regression, and gaussian process.

Mathematics

Kernel method

Given a feature map ϕ : R d → V {\textstyle \phi :\mathbb {R} ^{d}\to V} , where V {\textstyle V} is a Hilbert space (more specifically, a reproducing kernel Hilbert space), the kernel trick replaces inner products in feature space ⟨ ϕ ( x i ) , ϕ ( x j ) ⟩ V {\displaystyle \langle \phi (x_{i}),\phi (x_{j})\rangle _{V}} by a kernel function k ( x i , x j ) : R d × R d → R {\displaystyle k(x_{i},x_{j}):\mathbb {R} ^{d}\times \mathbb {R} ^{d}\to \mathbb {R} } Kernel methods replaces linear operations in high-dimensional space by operations on the kernel matrix: K X := [ k ( x i , x j ) ] i , j ∈ 1 : N {\displaystyle K_{X}:=[k(x_{i},x_{j})]_{i,j\in 1:N}} where N {\textstyle N} is the number of data points.

Random kernel method The problem with kernel methods is that the kernel matrix K X {\textstyle K_{X}} has size N × N {\textstyle N\times N} . This becomes computationally infeasible when N {\textstyle N} reaches the order of a million. The random kernel method replaces the kernel function k {\textstyle k} by an inner product in low-dimensional feature space R D {\textstyle \mathbb {R} ^{D}} : k ( x , y ) ≈ ⟨ z ( x ) , z ( y ) ⟩ {\displaystyle k(x,y)\approx \langle z(x),z(y)\rangle } where z {\textstyle z} is a randomly sampled feature map z : R d → R D {\textstyle z:\mathbb {R} ^{d}\to \mathbb {R} ^{D}} . This converts kernel linear regression into linear regression in feature space, kernel SVM into SVM in feature space, etc. Since we have K X ≈ Z X T Z X {\displaystyle K_{X}\approx Z_{X}^{T}Z_{X}} where Z X = [ z ( x 1 ) , … , z ( x N ) ] {\displaystyle Z_{X}=[z(x_{1}),\dots ,z(x_{N})]} , these methods no longer involve matrices of size O ( N 2 ) {\textstyle O(N^{2})} , but only random feature matrices of size O ( D N ) {\textstyle O(DN)} .

Random Fourier feature

Radial basis function kernel

The radial basis function (RBF) kernel on two samples x i , x j ∈ R d {\displaystyle x_{i},x_{j}\in \mathbb {R} ^{d}} is defined as

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Random feature

Start with the simplest possible case. Write down what Random feature 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 Random feature 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 Random feature 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 Random feature

In research: Random feature 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 Random feature 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: Random feature is common in secondary-school and first-year university syllabi. It links to neighbouring topics Kernel methods for machine learning, Machine learning, Monte Carlo methods, so understanding it makes those chapters shorter.
In everyday life: Look for Random feature 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 Random feature in 20 minutes

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

Frequently asked questions

What is Random feature in simple terms?

Random features (RF) are a technique used in machine learning to approximate kernel methods, introduced by Ali Rahimi and Ben Recht in their 2007 paper "Random Features for Large-Scale Kernel Machines", and extended by. RF uses a Monte Carlo approximation to kernel functions by randomly sampled fea…

Why does Random feature 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 Random feature?

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 Random feature.