science

Non-negative least squares

Non-negative least squares is a 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 Non-negative least squares rather than just read about it. In short: In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix A and a (column) vector of response variables y, the goal is to find a r g m i n x ⁡ ‖ A x − y ‖ 2 2 {\displaystyle \operatorname {arg\,min} \limits _{\mathbf {x} }\|\mathbf {Ax} -\mathbf {y} \|_{2}^{2}} subjec…

Key takeaways

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

Reference excerpt

In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix A and a (column) vector of response variables y, the goal is to find

a r g m i n x ⁡ ‖ A x − y ‖ 2 2 {\displaystyle \operatorname {arg\,min} \limits _{\mathbf {x} }\|\mathbf {Ax} -\mathbf {y} \|_{2}^{2}} subject to x ≥ 0. Here x ≥ 0 means that each component of the vector x should be non-negative, and ‖·‖2 denotes the Euclidean norm. Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC and non-negative matrix/tensor factorization. The latter can be considered a generalization of NNLS. Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds αi ≤ xi ≤ βi.

Quadratic programming version The NNLS problem is equivalent to a quadratic programming problem

a r g m i n x ≥ 0 ⁡ ( 1 2 x T Q x + c T x ) , {\displaystyle \operatorname {arg\,min} \limits _{\mathbf {x\geq 0} }\left({\frac {1}{2}}\mathbf {x} ^{\mathsf {T}}\mathbf {Q} \mathbf {x} +\mathbf {c} ^{\mathsf {T}}\mathbf {x} \right),}

where Q = ATA and c = −AT y. This problem is convex, as Q is positive semidefinite and the non-negativity constraints form a convex feasible set.

Algorithms The first widely used algorithm for solving this problem is an active-set method published by Lawson and Hanson in their 1974 book Solving Least Squares Problems. In pseudocode, this algorithm looks as follows:

This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse ((AP)T AP)−1. Variants of this algorithm are available in MATLAB as the routine lsqnonneg and in SciPy as optimize.nnls. Many improved algorithms have been suggested since 1974. Fast NNLS (FNNLS) is an optimized version of the Lawson–Hanson algorithm. Other algorithms include variants of Landweber's gradient descent method, coordinate-wise optimization based on the quadratic programming problem above, and an active set method called TNT-NN.

See also M-matrix Perron–Frobenius theorem

References

Worked examples

Example 1 — a first encounter with Non-negative least squares

Start with the simplest possible case. Write down what Non-negative least squares claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In 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 Non-negative least squares 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 Non-negative least squares 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 Non-negative least squares

In research: Non-negative least squares appears in 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 Non-negative least squares 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: Non-negative least squares is common in secondary-school and first-year university syllabi. It links to neighbouring topics Least squares, so understanding it makes those chapters shorter.
In everyday life: Look for Non-negative least squares 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 Non-negative least squares in 20 minutes

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

Frequently asked questions

What is Non-negative least squares in simple terms?

Why does Non-negative least squares matter?

Because it connects several 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 Non-negative least squares?

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 Non-negative least squares.