computer science

Low-rank matrix approximations

Low-rank matrix approximations is a computer science 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 Low-rank matrix approximations rather than just read about it. In short: Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. Kernel methods (for instance, support vector machines or Gaussian processes) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane.

Key takeaways

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

Reference excerpt

Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. Kernel methods (for instance, support vector machines or Gaussian processes) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane. In the kernel method the data is represented in a kernel matrix (or, Gram matrix). Many algorithms can solve machine learning problems using the kernel matrix. The main problem of kernel method is its high computational cost associated with kernel matrices. The cost is at least quadratic in the number of training data points, but most kernel methods include computation of matrix inversion or eigenvalue decomposition and the cost becomes cubic in the number of training data. Large training sets cause large storage and computational costs. While low rank decomposition methods (Cholesky decomposition) reduce this cost, they still require computing the kernel matrix. One of the approaches to deal with this problem is low-rank matrix approximations. The most popular examples of them are the Nyström approximation and randomized feature maps approximation methods. Both of them have been successfully applied to efficient kernel learning.

Nyström approximation Kernel methods become computationally unfeasible when the number of points D {\displaystyle D} is so large such that the kernel matrix K {\displaystyle K} cannot be stored in memory. If D {\displaystyle D} is the number of training examples, the storage and computational cost required to find the solution of the problem using general kernel method is O ( D 2 ) {\displaystyle O(D^{2})} and O ( D 3 ) {\displaystyle O(D^{3})} respectively. The Nyström approximation can allow a significant speed-up of the computations. This speed-up is achieved by using, instead of the kernel matrix, its approximation K ~ {\displaystyle {\tilde {K}}} of rank d {\displaystyle d} . An advantage of the method is that it is not necessary to compute or store the whole kernel matrix, but only a submatrix of size d × D {\displaystyle d\times D} . It reduces the storage and complexity requirements to O ( D d ) {\displaystyle O(Dd)} and O ( D d 2 ) {\displaystyle O(Dd^{2})} respectively. The method is named "Nyström approximation" because it can be interpreted as a case of the Nyström method from integral equation theory.

Kernel approximation

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Worked examples

Example 1 — a first encounter with Low-rank matrix approximations

Start with the simplest possible case. Write down what Low-rank matrix approximations 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 Low-rank matrix approximations 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 Low-rank matrix approximations 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 Low-rank matrix approximations

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

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

Frequently asked questions

What is Low-rank matrix approximations in simple terms?

Why does Low-rank matrix approximations 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 Low-rank matrix approximations?

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 Low-rank matrix approximations.