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# ----------------------------------------------------------------------------
# - Open3D: www.open3d.org -
# ----------------------------------------------------------------------------
# Copyright (c) 2018-2024 www.open3d.org
# SPDX-License-Identifier: MIT
# ----------------------------------------------------------------------------
import numpy as np
from collections import OrderedDict
def compute_jacobian_finite_differences(x0, fn, epsilon):
"""Computes the Jacobian using finite differences
x0: The positions at which to compute J.
fn: A function of the form fn(x) which returns a single numpy array.
epsilon: A scalar or an array that can be broadcasted to the same
shape as x0.
"""
dtype = x0.dtype
y0 = fn(x0)
h = np.zeros_like(x0)
J = np.zeros((x0.size, y0.size), dtype=dtype)
epsilon_arr = np.broadcast_to(epsilon, x0.shape)
for i in range(x0.size):
eps = epsilon_arr.flat[i]
h.flat[i] = eps
J[i, :] = ((fn(x0 + h) - y0) / eps).flat
h.flat[i] = 0
return J
def compute_jacobian_analytical(x0, y_shape, fn_grad, y_bp=None):
"""Computes the analytical Jacobian
x0: The position at which to compute J.
y_shape: The shape of the backpropagated value, i.e. the shape of
the output of the corresponding function 'fn'.
fn_grad: The gradient of the original function with the form
x_grad = fn_grad(y_bp, x0) where 'y_bp' is the backpropagated
value and 'x0' is the original input to 'fn'. The output of
the function is the gradient of x wrt to y.
y_bp: Optional array with custom values for individually scaling
the gradients.
"""
dtype = x0.dtype
y_size = 1
for k in y_shape:
y_size *= k
J = np.zeros((x0.size, y_size), dtype=dtype)
y = np.zeros(y_shape, dtype=dtype)
y_bp_arr = np.broadcast_to(y_bp, y_shape) if not y_bp is None else np.ones(
y_shape, dtype=dtype)
for j in range(y_size):
y.flat[j] = y_bp_arr.flat[j]
J[:, j] = fn_grad(y, x0).flat
y.flat[j] = 0
return J
def check_gradients(x0,
fn,
fn_grad,
epsilon=1e-6,
rtol=1e-3,
atol=1e-5,
debug_outputs=OrderedDict()):
"""Checks if the numerical and analytical gradients are compatible for a function 'fn'
x0: The position at which to compute the gradients.
fn: A function of the form fn(x) which returns a single numpy array.
fn_grad: The gradient of the original function with the form
x_grad = fn_grad(y_bp, x0) where 'y_bp' is the backpropagated
value and 'x0' is the original input to 'fn'. The output of
the function is the gradient of x wrt to y.
epsilon: A scalar or an array that can be broadcasted to the same
shape as x0. This is used for computing the numerical Jacobian
rtol: The relative tolerance parameter used in numpy.allclose()
atol: The absolute tolerance parameter used in numpy.allclose()
debug_outputs: Output variable which stores additional outputs useful for
debugging in a dictionary.
"""
dtype = x0.dtype
y = fn(x0) # compute y to get the shape
grad = fn_grad(np.zeros(y.shape, dtype=dtype), x0)
grad_shape_correct = x0.shape == grad.shape
if not grad_shape_correct:
print(
'The shape of the gradient [{0}] does not match the shape of "x0" [{1}].'
.format(grad.shape, x0.shape))
zero_grad = np.count_nonzero(grad) == 0
if not zero_grad:
print('The gradient is not zero for a zero backprop vector.')
ana_J = compute_jacobian_analytical(x0, y.shape, fn_grad)
ana_J2 = compute_jacobian_analytical(x0, y.shape, fn_grad,
2 * np.ones(y.shape, dtype=x0.dtype))
num_J = compute_jacobian_finite_differences(x0, fn, epsilon)
does_scale = np.allclose(0.5 * ana_J2, ana_J, rtol, atol)
isclose = np.allclose(ana_J, num_J, rtol, atol)
ana_J_iszero = np.all(ana_J == 0)
if ana_J_iszero and not np.allclose(num_J, np.zeros_like(num_J), rtol,
atol):
print(
'The values of the analytical Jacobian are all zero but the values of the numerical Jacobian are not.'
)
elif not does_scale:
print(
'The gradients do not scale with respect to the backpropagated values.'
)
if not isclose:
print('The gradients are not close to the numerical Jacobian.')
debug_outputs.update(
OrderedDict([
('isclose', isclose),
('does_scale', does_scale),
('ana_J_iszero', ana_J_iszero),
('grad_shape_correct', grad_shape_correct),
('zero_grad', zero_grad),
('ana_J', ana_J),
('num_J', num_J),
('absdiff', np.abs(ana_J - num_J)),
]))
result = isclose and does_scale
return result
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