Non-negative least squares

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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]

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] The latter can be considered a generalization of NNLS.^[1]

Quadratic programming version

The NNLS problem is equivalent to a quadratic programming problem

,

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.^[4]

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 : matrix of shape (m, n)
y : vector of length m
tol : tolerance for the stopping criterion

# Initialization
P ← ∅
R ← {1, ..., n}
x ← zero-vector of length n
w ← Aᵀ(y − Ax)

while R ≠ ∅ and max(w) > tol
    j ← index of max(w) in w
    add j to P
    remove j from R
    # Aᴾ is A restricted to the variables included in P
    sᴾ ← ((Aᴾ)ᵀ Aᴾ)⁻¹ (Aᴾ)ᵀy
    while min(sᴾ) ≤ 0
        α ← min(xᵢ / (xᵢ - sᵢ) for i in P, sᵢ ≤ 0)
        x ← x + α(s - x)
        Move to R all indexes j in P such that xj = 0
        sᴾ ← ((Aᴾ)ᵀ Aᴾ)⁻¹ (Aᴾ)ᵀy
        sᴿ ← 0
    x ← s
    w ← Aᵀ(y − Ax)

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.^[6]

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^[7] and coordinate-wise optimization based on the quadratic programming problem above.^[4] Lawson and Hanson themselves have generalized it to handle general bounded-variable least squares (BLVS) problems, with upper and lower bounds αᵢ ≤ xᵢ ≤ βᵢ.^[5]:291

References

See also

• M-matrix
• Perron–Frobenius theorem