Least-squares support vector machine

Key takeaways

Least-squares support vector machine 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 Least-squares support vector machine to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Least-squares support vector machine from memory before moving on to harder problems.

Reference excerpt

Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods.

From support-vector machine to least-squares support-vector machine Given a training set { x i , y i } i = 1 N {\displaystyle \{x_{i},y_{i}\}_{i=1}^{N}} with input data x i ∈ R n {\displaystyle x_{i}\in \mathbb {R} ^{n}} and corresponding binary class labels y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} , the SVM classifier, according to Vapnik's original formulation, satisfies the following conditions:

{ w T ϕ ( x i ) + b ≥ 1 , if y i = + 1 , w T ϕ ( x i ) + b ≤ − 1 , if y i = − 1 , {\displaystyle {\begin{cases}w^{T}\phi (x_{i})+b\geq 1,&{\text{if }}\quad y_{i}=+1,\\w^{T}\phi (x_{i})+b\leq -1,&{\text{if }}\quad y_{i}=-1,\end{cases}}}

which is equivalent to

y i [ w T ϕ ( x i ) + b ] ≥ 1 , i = 1 , … , N , {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1,\quad i=1,\ldots ,N,}

where ϕ ( x ) {\displaystyle \phi (x)} is the nonlinear map from original space to the high- or infinite-dimensional space.

Inseparable data In case such a separating hyperplane does not exist, we introduce so-called slack variables ξ i {\displaystyle \xi _{i}} such that

{ y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N . {\displaystyle {\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N.\end{cases}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Least-squares support vector machine

Start with the simplest possible case. Write down what Least-squares support vector machine 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 Least-squares support vector machine 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 Least-squares support vector machine 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 Least-squares support vector machine

In research: Least-squares support vector machine 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 Least-squares support vector machine 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: Least-squares support vector machine is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1999 in artificial intelligence, Classification algorithms, Kernel methods for machine learning, so understanding it makes those chapters shorter.
In everyday life: Look for Least-squares support vector machine 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 Least-squares support vector machine in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Least-squares support vector machine 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 Least-squares support vector machine out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Least-squares support vector machine in simple terms?

Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regressi…

Why does Least-squares support vector machine 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 Least-squares support vector machine?

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 Least-squares support vector machine.