You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
735 lines
20 KiB
Python
735 lines
20 KiB
Python
"""Functions used by least-squares algorithms."""
|
|
from math import copysign
|
|
|
|
import numpy as np
|
|
from numpy.linalg import norm
|
|
|
|
from scipy.linalg import cho_factor, cho_solve, LinAlgError
|
|
from scipy.sparse import issparse
|
|
from scipy.sparse.linalg import LinearOperator, aslinearoperator
|
|
|
|
|
|
EPS = np.finfo(float).eps
|
|
|
|
|
|
# Functions related to a trust-region problem.
|
|
|
|
|
|
def intersect_trust_region(x, s, Delta):
|
|
"""Find the intersection of a line with the boundary of a trust region.
|
|
|
|
This function solves the quadratic equation with respect to t
|
|
||(x + s*t)||**2 = Delta**2.
|
|
|
|
Returns
|
|
-------
|
|
t_neg, t_pos : tuple of float
|
|
Negative and positive roots.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
If `s` is zero or `x` is not within the trust region.
|
|
"""
|
|
a = np.dot(s, s)
|
|
if a == 0:
|
|
raise ValueError("`s` is zero.")
|
|
|
|
b = np.dot(x, s)
|
|
|
|
c = np.dot(x, x) - Delta**2
|
|
if c > 0:
|
|
raise ValueError("`x` is not within the trust region.")
|
|
|
|
d = np.sqrt(b*b - a*c) # Root from one fourth of the discriminant.
|
|
|
|
# Computations below avoid loss of significance, see "Numerical Recipes".
|
|
q = -(b + copysign(d, b))
|
|
t1 = q / a
|
|
t2 = c / q
|
|
|
|
if t1 < t2:
|
|
return t1, t2
|
|
else:
|
|
return t2, t1
|
|
|
|
|
|
def solve_lsq_trust_region(n, m, uf, s, V, Delta, initial_alpha=None,
|
|
rtol=0.01, max_iter=10):
|
|
"""Solve a trust-region problem arising in least-squares minimization.
|
|
|
|
This function implements a method described by J. J. More [1]_ and used
|
|
in MINPACK, but it relies on a single SVD of Jacobian instead of series
|
|
of Cholesky decompositions. Before running this function, compute:
|
|
``U, s, VT = svd(J, full_matrices=False)``.
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
Number of variables.
|
|
m : int
|
|
Number of residuals.
|
|
uf : ndarray
|
|
Computed as U.T.dot(f).
|
|
s : ndarray
|
|
Singular values of J.
|
|
V : ndarray
|
|
Transpose of VT.
|
|
Delta : float
|
|
Radius of a trust region.
|
|
initial_alpha : float, optional
|
|
Initial guess for alpha, which might be available from a previous
|
|
iteration. If None, determined automatically.
|
|
rtol : float, optional
|
|
Stopping tolerance for the root-finding procedure. Namely, the
|
|
solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``.
|
|
max_iter : int, optional
|
|
Maximum allowed number of iterations for the root-finding procedure.
|
|
|
|
Returns
|
|
-------
|
|
p : ndarray, shape (n,)
|
|
Found solution of a trust-region problem.
|
|
alpha : float
|
|
Positive value such that (J.T*J + alpha*I)*p = -J.T*f.
|
|
Sometimes called Levenberg-Marquardt parameter.
|
|
n_iter : int
|
|
Number of iterations made by root-finding procedure. Zero means
|
|
that Gauss-Newton step was selected as the solution.
|
|
|
|
References
|
|
----------
|
|
.. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation
|
|
and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes
|
|
in Mathematics 630, Springer Verlag, pp. 105-116, 1977.
|
|
"""
|
|
def phi_and_derivative(alpha, suf, s, Delta):
|
|
"""Function of which to find zero.
|
|
|
|
It is defined as "norm of regularized (by alpha) least-squares
|
|
solution minus `Delta`". Refer to [1]_.
|
|
"""
|
|
denom = s**2 + alpha
|
|
p_norm = norm(suf / denom)
|
|
phi = p_norm - Delta
|
|
phi_prime = -np.sum(suf ** 2 / denom**3) / p_norm
|
|
return phi, phi_prime
|
|
|
|
suf = s * uf
|
|
|
|
# Check if J has full rank and try Gauss-Newton step.
|
|
if m >= n:
|
|
threshold = EPS * m * s[0]
|
|
full_rank = s[-1] > threshold
|
|
else:
|
|
full_rank = False
|
|
|
|
if full_rank:
|
|
p = -V.dot(uf / s)
|
|
if norm(p) <= Delta:
|
|
return p, 0.0, 0
|
|
|
|
alpha_upper = norm(suf) / Delta
|
|
|
|
if full_rank:
|
|
phi, phi_prime = phi_and_derivative(0.0, suf, s, Delta)
|
|
alpha_lower = -phi / phi_prime
|
|
else:
|
|
alpha_lower = 0.0
|
|
|
|
if initial_alpha is None or not full_rank and initial_alpha == 0:
|
|
alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5)
|
|
else:
|
|
alpha = initial_alpha
|
|
|
|
for it in range(max_iter):
|
|
if alpha < alpha_lower or alpha > alpha_upper:
|
|
alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5)
|
|
|
|
phi, phi_prime = phi_and_derivative(alpha, suf, s, Delta)
|
|
|
|
if phi < 0:
|
|
alpha_upper = alpha
|
|
|
|
ratio = phi / phi_prime
|
|
alpha_lower = max(alpha_lower, alpha - ratio)
|
|
alpha -= (phi + Delta) * ratio / Delta
|
|
|
|
if np.abs(phi) < rtol * Delta:
|
|
break
|
|
|
|
p = -V.dot(suf / (s**2 + alpha))
|
|
|
|
# Make the norm of p equal to Delta, p is changed only slightly during
|
|
# this. It is done to prevent p lie outside the trust region (which can
|
|
# cause problems later).
|
|
p *= Delta / norm(p)
|
|
|
|
return p, alpha, it + 1
|
|
|
|
|
|
def solve_trust_region_2d(B, g, Delta):
|
|
"""Solve a general trust-region problem in 2 dimensions.
|
|
|
|
The problem is reformulated as a 4th order algebraic equation,
|
|
the solution of which is found by numpy.roots.
|
|
|
|
Parameters
|
|
----------
|
|
B : ndarray, shape (2, 2)
|
|
Symmetric matrix, defines a quadratic term of the function.
|
|
g : ndarray, shape (2,)
|
|
Defines a linear term of the function.
|
|
Delta : float
|
|
Radius of a trust region.
|
|
|
|
Returns
|
|
-------
|
|
p : ndarray, shape (2,)
|
|
Found solution.
|
|
newton_step : bool
|
|
Whether the returned solution is the Newton step which lies within
|
|
the trust region.
|
|
"""
|
|
try:
|
|
R, lower = cho_factor(B)
|
|
p = -cho_solve((R, lower), g)
|
|
if np.dot(p, p) <= Delta**2:
|
|
return p, True
|
|
except LinAlgError:
|
|
pass
|
|
|
|
a = B[0, 0] * Delta**2
|
|
b = B[0, 1] * Delta**2
|
|
c = B[1, 1] * Delta**2
|
|
|
|
d = g[0] * Delta
|
|
f = g[1] * Delta
|
|
|
|
coeffs = np.array(
|
|
[-b + d, 2 * (a - c + f), 6 * b, 2 * (-a + c + f), -b - d])
|
|
t = np.roots(coeffs) # Can handle leading zeros.
|
|
t = np.real(t[np.isreal(t)])
|
|
|
|
p = Delta * np.vstack((2 * t / (1 + t**2), (1 - t**2) / (1 + t**2)))
|
|
value = 0.5 * np.sum(p * B.dot(p), axis=0) + np.dot(g, p)
|
|
i = np.argmin(value)
|
|
p = p[:, i]
|
|
|
|
return p, False
|
|
|
|
|
|
def update_tr_radius(Delta, actual_reduction, predicted_reduction,
|
|
step_norm, bound_hit):
|
|
"""Update the radius of a trust region based on the cost reduction.
|
|
|
|
Returns
|
|
-------
|
|
Delta : float
|
|
New radius.
|
|
ratio : float
|
|
Ratio between actual and predicted reductions.
|
|
"""
|
|
if predicted_reduction > 0:
|
|
ratio = actual_reduction / predicted_reduction
|
|
elif predicted_reduction == actual_reduction == 0:
|
|
ratio = 1
|
|
else:
|
|
ratio = 0
|
|
|
|
if ratio < 0.25:
|
|
Delta = 0.25 * step_norm
|
|
elif ratio > 0.75 and bound_hit:
|
|
Delta *= 2.0
|
|
|
|
return Delta, ratio
|
|
|
|
|
|
# Construction and minimization of quadratic functions.
|
|
|
|
|
|
def build_quadratic_1d(J, g, s, diag=None, s0=None):
|
|
"""Parameterize a multivariate quadratic function along a line.
|
|
|
|
The resulting univariate quadratic function is given as follows:
|
|
::
|
|
f(t) = 0.5 * (s0 + s*t).T * (J.T*J + diag) * (s0 + s*t) +
|
|
g.T * (s0 + s*t)
|
|
|
|
Parameters
|
|
----------
|
|
J : ndarray, sparse matrix or LinearOperator shape (m, n)
|
|
Jacobian matrix, affects the quadratic term.
|
|
g : ndarray, shape (n,)
|
|
Gradient, defines the linear term.
|
|
s : ndarray, shape (n,)
|
|
Direction vector of a line.
|
|
diag : None or ndarray with shape (n,), optional
|
|
Addition diagonal part, affects the quadratic term.
|
|
If None, assumed to be 0.
|
|
s0 : None or ndarray with shape (n,), optional
|
|
Initial point. If None, assumed to be 0.
|
|
|
|
Returns
|
|
-------
|
|
a : float
|
|
Coefficient for t**2.
|
|
b : float
|
|
Coefficient for t.
|
|
c : float
|
|
Free term. Returned only if `s0` is provided.
|
|
"""
|
|
v = J.dot(s)
|
|
a = np.dot(v, v)
|
|
if diag is not None:
|
|
a += np.dot(s * diag, s)
|
|
a *= 0.5
|
|
|
|
b = np.dot(g, s)
|
|
|
|
if s0 is not None:
|
|
u = J.dot(s0)
|
|
b += np.dot(u, v)
|
|
c = 0.5 * np.dot(u, u) + np.dot(g, s0)
|
|
if diag is not None:
|
|
b += np.dot(s0 * diag, s)
|
|
c += 0.5 * np.dot(s0 * diag, s0)
|
|
return a, b, c
|
|
else:
|
|
return a, b
|
|
|
|
|
|
def minimize_quadratic_1d(a, b, lb, ub, c=0):
|
|
"""Minimize a 1-D quadratic function subject to bounds.
|
|
|
|
The free term `c` is 0 by default. Bounds must be finite.
|
|
|
|
Returns
|
|
-------
|
|
t : float
|
|
Minimum point.
|
|
y : float
|
|
Minimum value.
|
|
"""
|
|
t = [lb, ub]
|
|
if a != 0:
|
|
extremum = -0.5 * b / a
|
|
if lb < extremum < ub:
|
|
t.append(extremum)
|
|
t = np.asarray(t)
|
|
y = t * (a * t + b) + c
|
|
min_index = np.argmin(y)
|
|
return t[min_index], y[min_index]
|
|
|
|
|
|
def evaluate_quadratic(J, g, s, diag=None):
|
|
"""Compute values of a quadratic function arising in least squares.
|
|
|
|
The function is 0.5 * s.T * (J.T * J + diag) * s + g.T * s.
|
|
|
|
Parameters
|
|
----------
|
|
J : ndarray, sparse matrix or LinearOperator, shape (m, n)
|
|
Jacobian matrix, affects the quadratic term.
|
|
g : ndarray, shape (n,)
|
|
Gradient, defines the linear term.
|
|
s : ndarray, shape (k, n) or (n,)
|
|
Array containing steps as rows.
|
|
diag : ndarray, shape (n,), optional
|
|
Addition diagonal part, affects the quadratic term.
|
|
If None, assumed to be 0.
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray with shape (k,) or float
|
|
Values of the function. If `s` was 2-D, then ndarray is
|
|
returned, otherwise, float is returned.
|
|
"""
|
|
if s.ndim == 1:
|
|
Js = J.dot(s)
|
|
q = np.dot(Js, Js)
|
|
if diag is not None:
|
|
q += np.dot(s * diag, s)
|
|
else:
|
|
Js = J.dot(s.T)
|
|
q = np.sum(Js**2, axis=0)
|
|
if diag is not None:
|
|
q += np.sum(diag * s**2, axis=1)
|
|
|
|
l = np.dot(s, g)
|
|
|
|
return 0.5 * q + l
|
|
|
|
|
|
# Utility functions to work with bound constraints.
|
|
|
|
|
|
def in_bounds(x, lb, ub):
|
|
"""Check if a point lies within bounds."""
|
|
return np.all((x >= lb) & (x <= ub))
|
|
|
|
|
|
def step_size_to_bound(x, s, lb, ub):
|
|
"""Compute a min_step size required to reach a bound.
|
|
|
|
The function computes a positive scalar t, such that x + s * t is on
|
|
the bound.
|
|
|
|
Returns
|
|
-------
|
|
step : float
|
|
Computed step. Non-negative value.
|
|
hits : ndarray of int with shape of x
|
|
Each element indicates whether a corresponding variable reaches the
|
|
bound:
|
|
|
|
* 0 - the bound was not hit.
|
|
* -1 - the lower bound was hit.
|
|
* 1 - the upper bound was hit.
|
|
"""
|
|
non_zero = np.nonzero(s)
|
|
s_non_zero = s[non_zero]
|
|
steps = np.empty_like(x)
|
|
steps.fill(np.inf)
|
|
with np.errstate(over='ignore'):
|
|
steps[non_zero] = np.maximum((lb - x)[non_zero] / s_non_zero,
|
|
(ub - x)[non_zero] / s_non_zero)
|
|
min_step = np.min(steps)
|
|
return min_step, np.equal(steps, min_step) * np.sign(s).astype(int)
|
|
|
|
|
|
def find_active_constraints(x, lb, ub, rtol=1e-10):
|
|
"""Determine which constraints are active in a given point.
|
|
|
|
The threshold is computed using `rtol` and the absolute value of the
|
|
closest bound.
|
|
|
|
Returns
|
|
-------
|
|
active : ndarray of int with shape of x
|
|
Each component shows whether the corresponding constraint is active:
|
|
|
|
* 0 - a constraint is not active.
|
|
* -1 - a lower bound is active.
|
|
* 1 - a upper bound is active.
|
|
"""
|
|
active = np.zeros_like(x, dtype=int)
|
|
|
|
if rtol == 0:
|
|
active[x <= lb] = -1
|
|
active[x >= ub] = 1
|
|
return active
|
|
|
|
lower_dist = x - lb
|
|
upper_dist = ub - x
|
|
|
|
lower_threshold = rtol * np.maximum(1, np.abs(lb))
|
|
upper_threshold = rtol * np.maximum(1, np.abs(ub))
|
|
|
|
lower_active = (np.isfinite(lb) &
|
|
(lower_dist <= np.minimum(upper_dist, lower_threshold)))
|
|
active[lower_active] = -1
|
|
|
|
upper_active = (np.isfinite(ub) &
|
|
(upper_dist <= np.minimum(lower_dist, upper_threshold)))
|
|
active[upper_active] = 1
|
|
|
|
return active
|
|
|
|
|
|
def make_strictly_feasible(x, lb, ub, rstep=1e-10):
|
|
"""Shift a point to the interior of a feasible region.
|
|
|
|
Each element of the returned vector is at least at a relative distance
|
|
`rstep` from the closest bound. If ``rstep=0`` then `np.nextafter` is used.
|
|
"""
|
|
x_new = x.copy()
|
|
|
|
active = find_active_constraints(x, lb, ub, rstep)
|
|
lower_mask = np.equal(active, -1)
|
|
upper_mask = np.equal(active, 1)
|
|
|
|
if rstep == 0:
|
|
x_new[lower_mask] = np.nextafter(lb[lower_mask], ub[lower_mask])
|
|
x_new[upper_mask] = np.nextafter(ub[upper_mask], lb[upper_mask])
|
|
else:
|
|
x_new[lower_mask] = (lb[lower_mask] +
|
|
rstep * np.maximum(1, np.abs(lb[lower_mask])))
|
|
x_new[upper_mask] = (ub[upper_mask] -
|
|
rstep * np.maximum(1, np.abs(ub[upper_mask])))
|
|
|
|
tight_bounds = (x_new < lb) | (x_new > ub)
|
|
x_new[tight_bounds] = 0.5 * (lb[tight_bounds] + ub[tight_bounds])
|
|
|
|
return x_new
|
|
|
|
|
|
def CL_scaling_vector(x, g, lb, ub):
|
|
"""Compute Coleman-Li scaling vector and its derivatives.
|
|
|
|
Components of a vector v are defined as follows:
|
|
::
|
|
| ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf
|
|
v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf
|
|
| 1, otherwise
|
|
|
|
According to this definition v[i] >= 0 for all i. It differs from the
|
|
definition in paper [1]_ (eq. (2.2)), where the absolute value of v is
|
|
used. Both definitions are equivalent down the line.
|
|
Derivatives of v with respect to x take value 1, -1 or 0 depending on a
|
|
case.
|
|
|
|
Returns
|
|
-------
|
|
v : ndarray with shape of x
|
|
Scaling vector.
|
|
dv : ndarray with shape of x
|
|
Derivatives of v[i] with respect to x[i], diagonal elements of v's
|
|
Jacobian.
|
|
|
|
References
|
|
----------
|
|
.. [1] M.A. Branch, T.F. Coleman, and Y. Li, "A Subspace, Interior,
|
|
and Conjugate Gradient Method for Large-Scale Bound-Constrained
|
|
Minimization Problems," SIAM Journal on Scientific Computing,
|
|
Vol. 21, Number 1, pp 1-23, 1999.
|
|
"""
|
|
v = np.ones_like(x)
|
|
dv = np.zeros_like(x)
|
|
|
|
mask = (g < 0) & np.isfinite(ub)
|
|
v[mask] = ub[mask] - x[mask]
|
|
dv[mask] = -1
|
|
|
|
mask = (g > 0) & np.isfinite(lb)
|
|
v[mask] = x[mask] - lb[mask]
|
|
dv[mask] = 1
|
|
|
|
return v, dv
|
|
|
|
|
|
def reflective_transformation(y, lb, ub):
|
|
"""Compute reflective transformation and its gradient."""
|
|
if in_bounds(y, lb, ub):
|
|
return y, np.ones_like(y)
|
|
|
|
lb_finite = np.isfinite(lb)
|
|
ub_finite = np.isfinite(ub)
|
|
|
|
x = y.copy()
|
|
g_negative = np.zeros_like(y, dtype=bool)
|
|
|
|
mask = lb_finite & ~ub_finite
|
|
x[mask] = np.maximum(y[mask], 2 * lb[mask] - y[mask])
|
|
g_negative[mask] = y[mask] < lb[mask]
|
|
|
|
mask = ~lb_finite & ub_finite
|
|
x[mask] = np.minimum(y[mask], 2 * ub[mask] - y[mask])
|
|
g_negative[mask] = y[mask] > ub[mask]
|
|
|
|
mask = lb_finite & ub_finite
|
|
d = ub - lb
|
|
t = np.remainder(y[mask] - lb[mask], 2 * d[mask])
|
|
x[mask] = lb[mask] + np.minimum(t, 2 * d[mask] - t)
|
|
g_negative[mask] = t > d[mask]
|
|
|
|
g = np.ones_like(y)
|
|
g[g_negative] = -1
|
|
|
|
return x, g
|
|
|
|
|
|
# Functions to display algorithm's progress.
|
|
|
|
|
|
def print_header_nonlinear():
|
|
print("{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}{5:^15}"
|
|
.format("Iteration", "Total nfev", "Cost", "Cost reduction",
|
|
"Step norm", "Optimality"))
|
|
|
|
|
|
def print_iteration_nonlinear(iteration, nfev, cost, cost_reduction,
|
|
step_norm, optimality):
|
|
if cost_reduction is None:
|
|
cost_reduction = " " * 15
|
|
else:
|
|
cost_reduction = "{0:^15.2e}".format(cost_reduction)
|
|
|
|
if step_norm is None:
|
|
step_norm = " " * 15
|
|
else:
|
|
step_norm = "{0:^15.2e}".format(step_norm)
|
|
|
|
print("{0:^15}{1:^15}{2:^15.4e}{3}{4}{5:^15.2e}"
|
|
.format(iteration, nfev, cost, cost_reduction,
|
|
step_norm, optimality))
|
|
|
|
|
|
def print_header_linear():
|
|
print("{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}"
|
|
.format("Iteration", "Cost", "Cost reduction", "Step norm",
|
|
"Optimality"))
|
|
|
|
|
|
def print_iteration_linear(iteration, cost, cost_reduction, step_norm,
|
|
optimality):
|
|
if cost_reduction is None:
|
|
cost_reduction = " " * 15
|
|
else:
|
|
cost_reduction = "{0:^15.2e}".format(cost_reduction)
|
|
|
|
if step_norm is None:
|
|
step_norm = " " * 15
|
|
else:
|
|
step_norm = "{0:^15.2e}".format(step_norm)
|
|
|
|
print("{0:^15}{1:^15.4e}{2}{3}{4:^15.2e}".format(
|
|
iteration, cost, cost_reduction, step_norm, optimality))
|
|
|
|
|
|
# Simple helper functions.
|
|
|
|
|
|
def compute_grad(J, f):
|
|
"""Compute gradient of the least-squares cost function."""
|
|
if isinstance(J, LinearOperator):
|
|
return J.rmatvec(f)
|
|
else:
|
|
return J.T.dot(f)
|
|
|
|
|
|
def compute_jac_scale(J, scale_inv_old=None):
|
|
"""Compute variables scale based on the Jacobian matrix."""
|
|
if issparse(J):
|
|
scale_inv = np.asarray(J.power(2).sum(axis=0)).ravel()**0.5
|
|
else:
|
|
scale_inv = np.sum(J**2, axis=0)**0.5
|
|
|
|
if scale_inv_old is None:
|
|
scale_inv[scale_inv == 0] = 1
|
|
else:
|
|
scale_inv = np.maximum(scale_inv, scale_inv_old)
|
|
|
|
return 1 / scale_inv, scale_inv
|
|
|
|
|
|
def left_multiplied_operator(J, d):
|
|
"""Return diag(d) J as LinearOperator."""
|
|
J = aslinearoperator(J)
|
|
|
|
def matvec(x):
|
|
return d * J.matvec(x)
|
|
|
|
def matmat(X):
|
|
return d[:, np.newaxis] * J.matmat(X)
|
|
|
|
def rmatvec(x):
|
|
return J.rmatvec(x.ravel() * d)
|
|
|
|
return LinearOperator(J.shape, matvec=matvec, matmat=matmat,
|
|
rmatvec=rmatvec)
|
|
|
|
|
|
def right_multiplied_operator(J, d):
|
|
"""Return J diag(d) as LinearOperator."""
|
|
J = aslinearoperator(J)
|
|
|
|
def matvec(x):
|
|
return J.matvec(np.ravel(x) * d)
|
|
|
|
def matmat(X):
|
|
return J.matmat(X * d[:, np.newaxis])
|
|
|
|
def rmatvec(x):
|
|
return d * J.rmatvec(x)
|
|
|
|
return LinearOperator(J.shape, matvec=matvec, matmat=matmat,
|
|
rmatvec=rmatvec)
|
|
|
|
|
|
def regularized_lsq_operator(J, diag):
|
|
"""Return a matrix arising in regularized least squares as LinearOperator.
|
|
|
|
The matrix is
|
|
[ J ]
|
|
[ D ]
|
|
where D is diagonal matrix with elements from `diag`.
|
|
"""
|
|
J = aslinearoperator(J)
|
|
m, n = J.shape
|
|
|
|
def matvec(x):
|
|
return np.hstack((J.matvec(x), diag * x))
|
|
|
|
def rmatvec(x):
|
|
x1 = x[:m]
|
|
x2 = x[m:]
|
|
return J.rmatvec(x1) + diag * x2
|
|
|
|
return LinearOperator((m + n, n), matvec=matvec, rmatvec=rmatvec)
|
|
|
|
|
|
def right_multiply(J, d, copy=True):
|
|
"""Compute J diag(d).
|
|
|
|
If `copy` is False, `J` is modified in place (unless being LinearOperator).
|
|
"""
|
|
if copy and not isinstance(J, LinearOperator):
|
|
J = J.copy()
|
|
|
|
if issparse(J):
|
|
J.data *= d.take(J.indices, mode='clip') # scikit-learn recipe.
|
|
elif isinstance(J, LinearOperator):
|
|
J = right_multiplied_operator(J, d)
|
|
else:
|
|
J *= d
|
|
|
|
return J
|
|
|
|
|
|
def left_multiply(J, d, copy=True):
|
|
"""Compute diag(d) J.
|
|
|
|
If `copy` is False, `J` is modified in place (unless being LinearOperator).
|
|
"""
|
|
if copy and not isinstance(J, LinearOperator):
|
|
J = J.copy()
|
|
|
|
if issparse(J):
|
|
J.data *= np.repeat(d, np.diff(J.indptr)) # scikit-learn recipe.
|
|
elif isinstance(J, LinearOperator):
|
|
J = left_multiplied_operator(J, d)
|
|
else:
|
|
J *= d[:, np.newaxis]
|
|
|
|
return J
|
|
|
|
|
|
def check_termination(dF, F, dx_norm, x_norm, ratio, ftol, xtol):
|
|
"""Check termination condition for nonlinear least squares."""
|
|
ftol_satisfied = dF < ftol * F and ratio > 0.25
|
|
xtol_satisfied = dx_norm < xtol * (xtol + x_norm)
|
|
|
|
if ftol_satisfied and xtol_satisfied:
|
|
return 4
|
|
elif ftol_satisfied:
|
|
return 2
|
|
elif xtol_satisfied:
|
|
return 3
|
|
else:
|
|
return None
|
|
|
|
|
|
def scale_for_robust_loss_function(J, f, rho):
|
|
"""Scale Jacobian and residuals for a robust loss function.
|
|
|
|
Arrays are modified in place.
|
|
"""
|
|
J_scale = rho[1] + 2 * rho[2] * f**2
|
|
J_scale[J_scale < EPS] = EPS
|
|
J_scale **= 0.5
|
|
|
|
f *= rho[1] / J_scale
|
|
|
|
return left_multiply(J, J_scale, copy=False), f
|