165 lines
5.4 KiB
Python
165 lines
5.4 KiB
Python
import numpy as np
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from scipy.linalg import solve, LinAlgWarning
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import warnings
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__all__ = ['nnls']
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def nnls(A, b, maxiter=None, *, atol=None):
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"""
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Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``.
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This problem, often called as NonNegative Least Squares, is a convex
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optimization problem with convex constraints. It typically arises when
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the ``x`` models quantities for which only nonnegative values are
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attainable; weight of ingredients, component costs and so on.
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Parameters
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----------
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A : (m, n) ndarray
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Coefficient array
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b : (m,) ndarray, float
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Right-hand side vector.
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maxiter: int, optional
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Maximum number of iterations, optional. Default value is ``3 * n``.
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atol: float
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Tolerance value used in the algorithm to assess closeness to zero in
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the projected residual ``(A.T @ (A x - b)`` entries. Increasing this
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value relaxes the solution constraints. A typical relaxation value can
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be selected as ``max(m, n) * np.linalg.norm(a, 1) * np.spacing(1.)``.
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This value is not set as default since the norm operation becomes
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expensive for large problems hence can be used only when necessary.
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Returns
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-------
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x : ndarray
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Solution vector.
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rnorm : float
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The 2-norm of the residual, ``|| Ax-b ||_2``.
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See Also
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--------
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lsq_linear : Linear least squares with bounds on the variables
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Notes
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-----
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The code is based on [2]_ which is an improved version of the classical
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algorithm of [1]_. It utilizes an active set method and solves the KKT
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(Karush-Kuhn-Tucker) conditions for the non-negative least squares problem.
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References
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----------
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.. [1] : Lawson C., Hanson R.J., "Solving Least Squares Problems", SIAM,
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1995, :doi:`10.1137/1.9781611971217`
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.. [2] : Bro, Rasmus and de Jong, Sijmen, "A Fast Non-Negativity-
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Constrained Least Squares Algorithm", Journal Of Chemometrics, 1997,
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:doi:`10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-L`
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.optimize import nnls
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...
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>>> A = np.array([[1, 0], [1, 0], [0, 1]])
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>>> b = np.array([2, 1, 1])
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>>> nnls(A, b)
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(array([1.5, 1. ]), 0.7071067811865475)
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>>> b = np.array([-1, -1, -1])
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>>> nnls(A, b)
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(array([0., 0.]), 1.7320508075688772)
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"""
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A = np.asarray_chkfinite(A)
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b = np.asarray_chkfinite(b)
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if len(A.shape) != 2:
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raise ValueError("Expected a two-dimensional array (matrix)" +
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f", but the shape of A is {A.shape}")
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if len(b.shape) != 1:
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raise ValueError("Expected a one-dimensional array (vector)" +
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f", but the shape of b is {b.shape}")
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m, n = A.shape
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if m != b.shape[0]:
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raise ValueError(
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"Incompatible dimensions. The first dimension of " +
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f"A is {m}, while the shape of b is {(b.shape[0], )}")
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x, rnorm, mode = _nnls(A, b, maxiter, tol=atol)
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if mode != 1:
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raise RuntimeError("Maximum number of iterations reached.")
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return x, rnorm
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def _nnls(A, b, maxiter=None, tol=None):
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"""
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This is a single RHS algorithm from ref [2] above. For multiple RHS
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support, the algorithm is given in :doi:`10.1002/cem.889`
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"""
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m, n = A.shape
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AtA = A.T @ A
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Atb = b @ A # Result is 1D - let NumPy figure it out
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if not maxiter:
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maxiter = 3*n
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if tol is None:
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tol = 10 * max(m, n) * np.spacing(1.)
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# Initialize vars
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x = np.zeros(n, dtype=np.float64)
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s = np.zeros(n, dtype=np.float64)
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# Inactive constraint switches
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P = np.zeros(n, dtype=bool)
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# Projected residual
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w = Atb.copy().astype(np.float64) # x=0. Skip (-AtA @ x) term
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# Overall iteration counter
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# Outer loop is not counted, inner iter is counted across outer spins
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iter = 0
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while (not P.all()) and (w[~P] > tol).any(): # B
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# Get the "most" active coeff index and move to inactive set
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k = np.argmax(w * (~P)) # B.2
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P[k] = True # B.3
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# Iteration solution
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s[:] = 0.
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# B.4
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with warnings.catch_warnings():
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warnings.filterwarnings('ignore', message='Ill-conditioned matrix',
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category=LinAlgWarning)
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s[P] = solve(AtA[np.ix_(P, P)], Atb[P], assume_a='sym', check_finite=False)
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# Inner loop
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while (iter < maxiter) and (s[P].min() < 0): # C.1
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iter += 1
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inds = P * (s < 0)
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alpha = (x[inds] / (x[inds] - s[inds])).min() # C.2
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x *= (1 - alpha)
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x += alpha*s
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P[x <= tol] = False
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with warnings.catch_warnings():
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warnings.filterwarnings('ignore', message='Ill-conditioned matrix',
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category=LinAlgWarning)
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s[P] = solve(AtA[np.ix_(P, P)], Atb[P], assume_a='sym',
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check_finite=False)
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s[~P] = 0 # C.6
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x[:] = s[:]
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w[:] = Atb - AtA @ x
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if iter == maxiter:
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# Typically following line should return
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# return x, np.linalg.norm(A@x - b), -1
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# however at the top level, -1 raises an exception wasting norm
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# Instead return dummy number 0.
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return x, 0., -1
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return x, np.linalg.norm(A@x - b), 1
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