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Non-negative least squares

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Title: Non-negative least squares  
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Non-negative least squares

In mathematical optimization, the problem of non-negative least squares (NNLS) is a constrained version of the 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]

\operatorname*{arg\,min}_\mathbf{x} \|\mathbf{Ax} - \mathbf{y}\|_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 (BLVS), with simultaneous upper and lower bounds αx ≤ β.[5]:291[6]

Quadratic programming version

The NNLS problem is equivalent to a quadratic programming problem

\operatorname*{arg\,min}_\mathbf{x \ge 0} \frac{1}{2} \mathbf{x}^\mathsf{T} \mathbf{Q}\mathbf{x} + \mathbf{c}^\mathsf{T} \mathbf{x},

where Q = AA and c = Ay. 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 t, 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 = Aᵀ(yAx)
  • Main loop: while R ≠ ∅ and max(w) > t,
    • Let j be the index of max(w) in w
    • Add j to P
    • Remove j from R
    • Let AP be A restricted to the variables included in P
    • Let SP = ((AP)ᵀ AP)-1 (AP)ᵀy
    • While min(SP ≤ 0):
      • Let α = min(xi/xi - si) for i in P where si ≤ 0
      • Set x to x + α(s - x)
      • Move to R all indices j in P such that xj = 0
      • Set sP = ((AP)ᵀ AP)-1 (AP)ᵀy
      • Set sR to zero
      • Set x to s
      • Set w to Aᵀ(yAx)

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.[1] Variants of this algorithm are available in Matlab as the routine lsqnonneg[1] and in SciPy as optimize.nnls.[8]

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[9] and coordinate-wise optimization based on the quadratic programming problem above.[7]

See also

References

  1. ^ a b c d e f
  2. ^ a b c
  3. ^
  4. ^
  5. ^ a b
  6. ^
  7. ^ a b
  8. ^
  9. ^
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