Non-negative least squares

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^[1]

${\displaystyle \operatorname {arg\,min} \limits _{\mathbf {x} }\|\mathbf {Ax} -\mathbf {y} \|_{2}^{2}}$ 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^[2] and non-negative matrix/tensor factorization.^[3][4] The latter can be considered a generalization of NNLS.^[1]

Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds α_i ≤ x_i ≤ β_i.^{[5]: 291 [6]}

Quadratic programming version

The NNLS problem is equivalent to a quadratic programming problem

${\displaystyle \operatorname {arg\,min} \limits _{\mathbf {x\geq 0} }\left({\frac {1}{2}}\mathbf {x} ^{\mathsf {T}}\mathbf {Q} \mathbf {x} +\mathbf {c} ^{\mathsf {T}}\mathbf {x} \right),}$

where Q = A^TA and c = −A^T y. This problem is convex, as Q is positive semidefinite and the non-negativity constraints form a convex feasible set.^[7]

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.^[5]: 291 In pseudocode, this algorithm looks as follows:^[1][2]

• Inputs:
  • a real-valued matrix A of dimension m × n,
  • a real-valued vector y of dimension m,
  • a real value ε, the tolerance for the stopping criterion.
• Initialize:
  • Set P = ∅.
  • Set R = {1, ..., n}.
  • Set x to an all-zero vector of dimension n.
  • Set w = A^T(y − Ax).
  • Let w^R denote the sub-vector with indexes from R
• Main loop: while R ≠ ∅ and max(w^R) > ε:
  • Let j in R be the index of max(w^R) in w.
  • Add j to P.
  • Remove j from R.
  • Let A^P be A restricted to the variables included in P.
  • Let s be vector of same length as x. Let s^P denote the sub-vector with indexes from P, and let s^R denote the sub-vector with indexes from R.
  • Set s^P = ((A^P)^T A^P)⁻¹ (A^P)^Ty
  • Set s^R to zero
  • While min(s^P) ≤ 0:
    • Let α = min ⁠x_i/x_i − s_i⁠ for i in P where s_i ≤ 0.
    • Set x to x + α(s − x).
    • Move to R all indices j in P such that x_j ≤ 0.
    • Set s^P = ((A^P)^T A^P)⁻¹ (A^P)^Ty
    • Set s^R to zero.
  • Set x to s.
  • Set w to A^T(y − Ax).
• Output: x

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 ((A^P)^T A^P)⁻¹.^[1] Variants of this algorithm are available in MATLAB as the routine lsqnonneg^[8][1] and in SciPy as optimize.nnls.^[9]

Many improved algorithms have been suggested since 1974.^[1] Fast NNLS (FNNLS) is an optimized version of the Lawson–Hanson algorithm.^[2] Other algorithms include variants of Landweber's gradient descent method^[10] and coordinate-wise optimization based on the quadratic programming problem above.^[7]

See also

• M-matrix
• Perron–Frobenius theorem

References

1. Chen, Donghui; Plemmons, Robert J. (2009). Nonnegativity constraints in numerical analysis. Symposium on the Birth of Numerical Analysis. CiteSeerX 10.1.1.157.9203.
2. Bro, Rasmus; De Jong, Sijmen (1997). "A fast non-negativity-constrained least squares algorithm". Journal of Chemometrics. 11 (5): 393. doi:10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-L.
3. Lin, Chih-Jen (2007). "Projected Gradient Methods for Nonnegative Matrix Factorization" (PDF). Neural Computation. 19 (10): 2756–2779. CiteSeerX 10.1.1.308.9135. doi:10.1162/neco.2007.19.10.2756. PMID 17716011.
4. Boutsidis, Christos; Drineas, Petros (2009). "Random projections for the nonnegative least-squares problem". Linear Algebra and Its Applications. 431 (5–7): 760–771. arXiv:0812.4547. doi:10.1016/j.laa.2009.03.026.
5. Lawson, Charles L.; Hanson, Richard J. (1995). "23. Linear Least Squares with Linear Inequality Constraints". Solving Least Squares Problems. SIAM. p. 161. doi:10.1137/1.9781611971217.ch23. ISBN 978-0-89871-356-5.
6. Stark, Philip B.; Parker, Robert L. (1995). "Bounded-variable least-squares: an algorithm and applications" (PDF). Computational Statistics. 10: 129.
7. Franc, Vojtěch; Hlaváč, Václav; Navara, Mirko (2005). "Sequential Coordinate-Wise Algorithm for the Non-negative Least Squares Problem". Computer Analysis of Images and Patterns. Lecture Notes in Computer Science. Vol. 3691. pp. 407–414. doi:10.1007/11556121_50. ISBN 978-3-540-28969-2.
8. "lsqnonneg". MATLAB Documentation. Retrieved October 28, 2022.
9. "scipy.optimize.nnls". SciPy v0.13.0 Reference Guide. Retrieved 25 January 2014.
10. Johansson, B. R.; Elfving, T.; Kozlov, V.; Censor, Y.; Forssén, P. E.; Granlund, G. S. (2006). "The application of an oblique-projected Landweber method to a model of supervised learning". Mathematical and Computer Modelling. 43 (7–8): 892. doi:10.1016/j.mcm.2005.12.010.