def test_force(a_tensor, b_tensor):
# a_eval, a_evec = np.linalg.eigh(a_tensor)
# b_eval, b_evec = np.linalg.eigh(b_tensor)
a_eval, a_evec = evp.dsyevv3(a_tensor)
b_eval, b_evec = evp.dsyevv3(b_tensor)
# print("ref w", a_eval)
# print("test w", evp.dsyevc3(a_tensor))
# print("ref v", a_evec)
# print("test v", evp.dsyevv3(a_tensor))
# assert 0
r = np.matmul(np.transpose(a_evec), b_evec)
I = np.eye(3)
rI = r * I # 3x3 -> 3x3
pos = np.sum(rI, axis=-1) # 3x3 -> 3
neg = -np.sum(rI, axis=-1) # 3x3 -> 3
acos_pos = np.arccos(pos) # 3 -> 3
acos_neg = np.arccos(neg) # 3 -> 3
a = np.amin([acos_pos, acos_neg], axis=0) # 2x3 -> 3
a2 = a * a # 3->3
l = np.sum(a2) # 3->1
# derivatives, start backprop
dl_da2 = np.ones(3) # 1 x 3
da2_da = 2 * a * np.eye(3) # 3 x 3
da_darg = np.stack(
[np.eye(3) * (acos_pos < acos_neg),
np.eye(3) * (acos_neg < acos_pos)])
darg_dpn = np.stack([
np.eye(3) * (-1 / np.sqrt(1 - pos * pos)),
np.eye(3) * (-1 / np.sqrt(1 - neg * neg))
])
dl_darg = np.matmul(np.matmul(dl_da2, da2_da), da_darg)
dpos = dl_darg[0] * (-1 / np.sqrt(1 - pos * pos))
dneg = dl_darg[1] * (-1 / np.sqrt(1 - neg * neg))
dneg = -dneg
dpn_dr = np.array([
[[1, 1, 1], [0, 0, 0], [0, 0, 0]],
[[0, 0, 0], [1, 1, 1], [0, 0, 0]],
[[0, 0, 0], [0, 0, 0], [1, 1, 1]],
])
# element wise
dr = (np.matmul(dpos, dpn_dr) + np.matmul(dneg, dpn_dr)) * np.eye(3)
dr_daevec = np.matmul(b_evec, dr.T)
dr_dbevec = np.matmul(a_evec, dr.T)
dl_datensor = grad_eigh(a_eval, a_evec, np.zeros_like(a_eval), dr_daevec)
dl_dbtensor = grad_eigh(b_eval, b_evec, np.zeros_like(b_eval), dr_dbevec)
return dl_datensor, dl_dbtensor
def pmi_u(r):
I = np.eye(3)
loss = []
for v, e in zip(r, I):
a_pos = np.arccos(np.sum(v * e)) # norm is always 1
a_neg = np.arccos(np.sum(-v * e)) # norm is always 1
a = np.amin([a_pos, a_neg])
loss.append(a * a)
return np.sum(loss)
def simplified_u(a_tensor, b_tensor):
a_eval, a_evec = np.linalg.eigh(a_tensor)
b_eval, b_evec = np.linalg.eigh(b_tensor)
r = np.matmul(np.transpose(a_evec), b_evec)
I = np.eye(3)
rI = r * I # 3x3 -> 3x3
pos = np.sum(rI, axis=-1)
neg = np.sum(-rI, axis=-1)
acos_pos = np.arccos(pos)
acos_neg = np.arccos(neg)
# [a,b,c]
# [d,e,f]
# -------
# [min(a,d), min(b,e), min(c,f)]
a = np.amin([acos_pos, acos_neg], axis=0)
return np.sum(a * a)
def _arccos(x, do_backprop):
if do_backprop:
# https://github.com/google/jax/issues/654
x = np.where(np.abs(x) >= 1, np.sign(x), x)
else:
x = np.clip(x, -1, 1)
return np.arccos(x)
def grassman_distance(y1, y2): """Grassman distance between subspaces spanned by Y1 and Y2.""" q1, _ = jnp.linalg.qr(y1) q2, _ = jnp.linalg.qr(y2) _, sigma, _ = jnp.linalg.svd(q1.T @ q2) sigma = jnp.round(sigma, decimals=6) return jnp.linalg.norm(jnp.arccos(sigma))
def angle(x):
r1 = x[0,:]-x[1,:]
r2 = x[2,:]-x[1,:]
# costheta = np.dot(r1,r2) / np.linalg.norm(r1) / np.linalg.norm(r2)
costheta = np.dot(r1,r2) / np.sqrt(np.dot(r1,r1) * np.dot(r2,r2))
theta = np.arccos(costheta)
return theta
def compute_grassman_distance(Y1, Y2):
"""Grassman distance between subspaces spanned by Y1 and Y2."""
Q1, _ = jnp.linalg.qr(Y1)
Q2, _ = jnp.linalg.qr(Y2)
_, sigma, _ = jnp.linalg.svd(Q1.T @ Q2)
sigma = jnp.round(sigma, decimals=6)
return jnp.linalg.norm(jnp.arccos(sigma))
def _von_mises_centered(key, concentration, shape, dtype):
# Cutoff from TensorFlow probability
# (https://github.com/tensorflow/probability/blob/f051e03dd3cc847d31061803c2b31c564562a993/tensorflow_probability/python/distributions/von_mises.py#L567-L570)
s_cutoff_map = {
jnp.dtype(jnp.float16): 1.8e-1,
jnp.dtype(jnp.float32): 2e-2,
jnp.dtype(jnp.float64): 1.2e-4,
}
s_cutoff = s_cutoff_map.get(dtype)
r = 1.0 + jnp.sqrt(1.0 + 4.0 * concentration**2)
rho = (r - jnp.sqrt(2.0 * r)) / (2.0 * concentration)
s_exact = (1.0 + rho**2) / (2.0 * rho)
s_approximate = 1.0 / concentration
s = jnp.where(concentration > s_cutoff, s_exact, s_approximate)
def cond_fn(*args):
""" check if all are done or reached max number of iterations """
i, _, done, _, _ = args[0]
return jnp.bitwise_and(i < 100, jnp.logical_not(jnp.all(done)))
def body_fn(*args):
i, key, done, _, w = args[0]
uni_ukey, uni_vkey, key = random.split(key, 3)
u = random.uniform(
key=uni_ukey,
shape=shape,
dtype=concentration.dtype,
minval=-1.0,
maxval=1.0,
)
z = jnp.cos(jnp.pi * u)
w = jnp.where(done, w,
(1.0 + s * z) / (s + z)) # Update where not done
y = concentration * (s - w)
v = random.uniform(key=uni_vkey,
shape=shape,
dtype=concentration.dtype)
accept = (y * (2.0 - y) >= v) | (jnp.log(y / v) + 1.0 >= y)
return i + 1, key, accept | done, u, w
init_done = jnp.zeros(shape, dtype=bool)
init_u = jnp.zeros(shape)
init_w = jnp.zeros(shape)
_, _, done, u, w = lax.while_loop(
cond_fun=cond_fn,
body_fun=body_fn,
init_val=(jnp.array(0), key, init_done, init_u, init_w),
)
return jnp.sign(u) * jnp.arccos(w)
def pmi_restraints(conf, params, box, lamb, a_idxs, b_idxs, masses,
angle_force, com_force):
a_com, a_tensor = inertia_tensor(conf[a_idxs], masses[a_idxs])
b_com, b_tensor = inertia_tensor(conf[b_idxs], masses[b_idxs])
a_eval, a_evec = np.linalg.eigh(a_tensor) # already sorted
b_eval, b_evec = np.linalg.eigh(b_tensor) # already sorted
r = np.matmul(np.transpose(a_evec), b_evec)
I = np.eye(3)
loss = []
for v, e in zip(r, I):
a_pos = np.arccos(np.sum(v * e)) # norm is always 1
a_neg = np.arccos(np.sum(-v * e)) # norm is always 1
a = np.amin([a_pos, a_neg])
loss.append(a * a)
def grassman_distance(Y1, Y2): # pylint: disable=invalid-name """Grassman distance between subspaces spanned by Y1 and Y2.""" Q1, _ = jnp.linalg.qr(Y1) # pylint: disable=invalid-name Q2, _ = jnp.linalg.qr(Y2) # pylint: disable=invalid-name _, sigma, _ = jnp.linalg.svd(Q1.T @ Q2) # sigma = jnp.clip(sigma, -1., 1.) sigma = jnp.round(sigma, decimals=6) return jnp.linalg.norm(jnp.arccos(sigma))
def cartesian_to_spherical(x):
r = jnp.sqrt(jnp.sum(x**2))
denominators = jnp.sqrt(jnp.cumsum(x[::-1]**2)[::-1])[:-1]
phi = jnp.arccos(x[:-1] / denominators)
last_value = jnp.where(x[-1] >= 0, phi[-1], 2 * jnp.pi - phi[-1])
phi = jax.ops.index_update(phi, -1, last_value)
return jnp.hstack([r, phi])
def inv_kin(self, target=None):
if target is None:
target = self.target
a = target[0]**2 + target[1]**2 - self.l[0]**2 - self.l[1]**2
b = 2 * self.l[0] * self.l[1]
q2 = np.arccos(a / b)
c = np.arctan2(target[1], target[0])
q1 = c - np.arctan2(self.l[1] * np.sin(q2),
(self.l[0] + self.l[1] * np.cos(q2)))
return np.array([q1, q2])
def jac_log(x: jnp.ndarray, y: jnp.ndarray) -> jnp.ndarray:
"""Computes the Jacobian of the logarithmic map on the sphere.
Args:
x: Origination points on the sphere.
y: Destination points on the sphere.
Returns:
d: The Jacobian of the logarithmic map.
"""
r = (x * y).sum(axis=-1, keepdims=True)
v = r * jnp.arccos(r) / jnp.power(1 - jnp.square(r), 1.5) - 1 / (1 - jnp.square(r))
a = v[..., jnp.newaxis] * (y - r * x)[..., jnp.newaxis] * x[..., jnp.newaxis, :]
acr = jnp.arccos(r)
b = (acr / jnp.sin(acr))[..., jnp.newaxis]
c = jnp.eye(x.shape[-1])[jnp.newaxis] - x[..., jnp.newaxis] * x[..., jnp.newaxis, :]
d = a + b * c
return d
def table_ambisonics_order_vs_rE(max_order=20):
"""Return a dataframe with rE as a function of order."""
order = np.arange(1, max_order + 1, dtype=np.int32)
rE3 = np.array(list(map(shelf.max_rE_3d, order)))
drE3 = np.append(np.nan, rE3[1:] - rE3[:-1])
rE2 = np.array(list(map(shelf.max_rE_2d, order)))
drE2 = np.append(np.nan, rE2[1:] - rE2[:-1])
df = pd.DataFrame(np.column_stack((
order,
rE2,
100 * drE2 / rE2,
2 * np.arccos(rE2) * 180 / π,
rE3,
100 * drE3 / rE3,
2 * np.arccos(rE3) * 180 / π,
)),
columns=('order', '2D', '% change', 'asw', '3D',
'% change', 'asw'))
return df
def harmonic_angle(conf, params, box, angle_idxs, param_idxs, cos_angles=True):
"""
Compute the harmonic bond energy given a collection of molecules.
This implements a harmonic angle potential: V(t) = k*(t - t0)^2 or V(t) = k*(cos(t)-cos(t0))^2
Parameters:
-----------
conf: shape [num_atoms, 3] np.array
atomic coordinates
params: shape [num_params,] np.array
unique parameters
box: shape [3, 3] np.array
periodic boundary vectors, if not None
angle_idxs: shape [num_angles, 3] np.array
each element (a, b, c) is a unique angle in the conformation. atom b is defined
to be the middle atom.
param_idxs: shape [num_angles, 2] np.array
each element (k_idx, t_idx) maps into params for angle constants and ideal angles
cos_angles: True (default)
if True, then this instead implements V(t) = k*(cos(t)-cos(t0))^2. This is far more
numerically stable when the angle is pi.
"""
ci = conf[angle_idxs[:, 0]]
cj = conf[angle_idxs[:, 1]]
ck = conf[angle_idxs[:, 2]]
kas = params[param_idxs[:, 0]]
a0s = params[param_idxs[:, 1]]
vij = delta_r(ci, cj, box)
vjk = delta_r(ck, cj, box)
top = np.sum(np.multiply(vij, vjk), -1)
bot = np.linalg.norm(vij, axis=-1) * np.linalg.norm(vjk, axis=-1)
tb = top / bot
# (ytz): we used the squared version so that we make this energy being strictly positive
if cos_angles:
energies = kas / 2 * np.power(tb - np.cos(a0s), 2)
else:
angle = np.arccos(tb)
energies = kas / 2 * np.power(angle - a0s, 2)
return np.sum(energies, -1) # reduce over all angles
def vector_angle(a, b):
"""
Find the angle between two vectors.
"""
a_mod = np.linalg.norm(a)
b_mod = np.linalg.norm(b)
if a.ndim == 2 & b.ndim == 2:
dot = np.einsum('ij,ik->i', a / a_mod, b / b_mod)
elif a.ndim == 1 & b.ndim == 1:
dot = np.dot(a / a_mod, b / b_mod)
else:
raise Exception('Input must have 1 or 2 dimensions.')
angle = np.arccos(dot)
return angle
def get_rotation_pytree(src: Any, dst: Any) -> Any:
"""
Takes two n-dimensional vectors/Pytree and returns an
nxn rotation matrix mapping cjax to dst.
Raises Value Error when unsuccessful.
"""
def __assert_rotation(R):
if R.ndim != 2:
print("R must be a matrix")
a, b = R.shape
if a != b:
print("R must be square")
if (not jnp.isclose(
jnp.abs(jnp.eye(a) - jnp.dot(R, R.T)).max(), 0.0, rtol=0.5)
) or (not jnp.isclose(
jnp.abs(jnp.eye(a) - jnp.dot(R.T, R)).max(), 0.0, rtol=0.5)):
print("R is not diagonal")
if not pytree_shape_array_equal(src, dst):
print("cjax and dst must be 1-dimensional arrays with the same shape.")
x = pytree_normalized(src)
y = pytree_normalized(dst)
n = len(dst)
# compute angle between x and y in their spanning space
theta = jnp.arccos(jnp.dot(
x, y)) # they are normalized so there is no denominator
if jnp.isclose(theta, 0):
print("x and y are co-linear")
# construct the 2d rotation matrix connecting x to y in their spanning space
R = jnp.array([[jnp.cos(theta), -jnp.sin(theta)],
[jnp.sin(theta), jnp.cos(theta)]])
__assert_rotation(R)
# get projections onto Span<x,y> and its orthogonal complement
u = x
v = pytree_normalized(pytree_sub(y, (jnp.dot(u, y) * u)))
P = jnp.outer(u, u.T) + jnp.outer(
v, v.T) # projection onto 2d space spanned by x and y
Q = jnp.eye(
n) - P # projection onto the orthogonal complement of Span<x,y>
# lift the rotation matrix into the n-dimensional space
uv = jnp.hstack((u[:, None], v[:, None]))
R = Q + jnp.dot(uv, jnp.dot(R, uv.T))
__assert_rotation(R)
if jnp.any(jnp.logical_not(jnp.isclose(jnp.dot(R, x), y, rtol=0.25))):
print("Rotation matrix did not work")
return R
def logarithmic(x: jnp.ndarray, y: jnp.ndarray) -> jnp.ndarray:
"""Computes the logarithmic map on the sphere.
Args:
x: Origination points on the sphere.
y: Destination points on the sphere.
Returns:
lg: The tangent vector in the tangent space of the origination point that
would produce the destination under the exponential map.
"""
xy = (x * y).sum(axis=-1, keepdims=True)
v = jnp.arccos(xy)
lg = v / jnp.sin(v) * (y - xy * x)
return lg
def _compute_angle(p0: jnp.ndarray, p1: jnp.ndarray,
p2: jnp.ndarray) -> jnp.float32:
"""Compute the angle centered at `p1` between the other two points."""
a = p1 - p0
da = np.linalg.norm(a)
b = p1 - p2
db = np.linalg.norm(b)
c = p0 - p2
dc = np.linalg.norm(c)
x = (da**2 + db**2 - dc**2) / (2.0 * da * db)
angle = jnp.arccos(x) * 180.0 / jnp.pi
return min(angle, 360 - angle)
def arccos(x): if isinstance(x, JaxArray): x = x.value return JaxArray(jnp.arccos(x))
def arccos(a: Numeric):
return jnp.arccos(a)
def harmonic_angle(conf,
params,
box,
lamb,
angle_idxs,
lamb_mult=None,
lamb_offset=None,
cos_angles=True):
"""
Compute the harmonic angle energy given a collection of molecules.
This implements a harmonic angle potential:
V(t) = k*(t - t0)^2
if cos_angles=False
or
V(t) = k*(cos(t)-cos(t0))^2
if cos_angles=True
Parameters:
-----------
conf: shape [num_atoms, 3] np.array
atomic coordinates
params: shape [num_params,] np.array
unique parameters
box: shape [3, 3] np.array
periodic boundary vectors, if not None
lamb: float
alchemical lambda parameter, linearly rescaled
lamb_mult: None, or broadcastable to angle_idxs.shape[0]
prefactor = (lamb_offset + lamb_mult * lamb)
lamb_offset: None, or broadcastable to angle_idxs.shape[0]
prefactor = (lamb_offset + lamb_mult * lamb)
angle_idxs: shape [num_angles, 3] np.array
each element (a, b, c) is a unique angle in the conformation. atom b is defined
to be the middle atom.
cos_angles: True (default)
if True, then this instead implements V(t) = k*(cos(t)-cos(t0))^2. This is far more
numerically stable when the angle is pi.
Notes:
------
* lamb argument unused
"""
if lamb_mult is None or lamb_offset is None or lamb is None:
assert lamb_mult is None
assert lamb_offset is None
prefactor = 1.0
else:
assert lamb_mult is not None
assert lamb_offset is not None
prefactor = lamb_offset + lamb_mult * lamb
ci = conf[angle_idxs[:, 0]]
cj = conf[angle_idxs[:, 1]]
ck = conf[angle_idxs[:, 2]]
kas = params[:, 0]
a0s = params[:, 1]
vij = ci - cj
vjk = ck - cj
top = np.sum(np.multiply(vij, vjk), -1)
bot = np.linalg.norm(vij, axis=-1) * np.linalg.norm(vjk, axis=-1)
tb = top / bot
# (ytz): we use the squared version so that the energy is strictly positive
if cos_angles:
energies = prefactor * kas / 2 * np.power(tb - np.cos(a0s), 2)
else:
angle = np.arccos(tb)
energies = prefactor * kas / 2 * np.power(angle - a0s, 2)
return np.sum(energies, -1) # reduce over all angles
def angle(P): return np.arccos(cosAngle(P))
def _asinc_naive(x):
return jnp.arccos(x) / jnp.sqrt(1 - x**2)
def get_index(x):
x = 2 * (grid.lower - x.flatten()) / grid.size + 1
idx = (grid.shape * np.arccos(x)) // np.pi
idx = np.nan_to_num(idx, nan=grid.shape)
return (*np.flip(np.uint32(idx)), )
def calc_angles(xyz: np.ndarray, struct_descr_dict):
r_ij = xyz[struct_descr_dict['angles'][:, 1]] - xyz[struct_descr_dict['angles'][:, 0]]
r_kj = xyz[struct_descr_dict['angles'][:, 1]] - xyz[struct_descr_dict['angles'][:, 2]]
cos = np.sum(r_ij * r_kj, axis=1) / (np.linalg.norm(r_ij, axis=1) * np.linalg.norm(r_kj, axis=1))
return np.arccos(np.clip(cos, -1, 1))
def safe_acos(t, eps=1e-8):
"""A safe version of arccos which avoids evaluating at -1 or 1."""
return jnp.arccos(jnp.clip(t, -1.0 + eps, 1.0 - eps))
def _top_k(input, k=1, sorted=True, name=None): # pylint: disable=unused-argument,redefined-builtin
raise NotImplementedError
# --- Begin Public Functions --------------------------------------------------
abs = utils.copy_docstring( # pylint: disable=redefined-builtin
tf.math.abs,
lambda x, name=None: np.abs(x))
accumulate_n = utils.copy_docstring(
tf.math.accumulate_n,
lambda inputs, shape=None, tensor_dtype=None, name=None: ( # pylint: disable=g-long-lambda
sum(map(np.array, inputs)).astype(utils.numpy_dtype(tensor_dtype))))
acos = utils.copy_docstring(tf.math.acos, lambda x, name=None: np.arccos(x))
acosh = utils.copy_docstring(tf.math.acosh, lambda x, name=None: np.arccosh(x))
add = utils.copy_docstring(tf.math.add, lambda x, y, name=None: np.add(x, y))
add_n = utils.copy_docstring(
tf.math.add_n, lambda inputs, name=None: sum(map(np.array, inputs)))
angle = utils.copy_docstring(tf.math.angle,
lambda input, name=None: np.angle(input))
argmax = utils.copy_docstring(
tf.math.argmax,
lambda input, axis=None, output_type=tf.int64, name=None: ( # pylint: disable=g-long-lambda
np.argmax(input, axis=0 if axis is None else _astuple(axis)).astype(
def _angle(pos: jnp.ndarray, indices: jnp.ndarray,
tvecs: jnp.ndarray) -> float:
dx1 = -(pos[indices[1]] - pos[indices[0]] + tvecs[0])
dx2 = pos[indices[2]] - pos[indices[1]] + tvecs[1]
return jnp.arccos(dx1 @ dx2 /
(jnp.linalg.norm(dx1) * jnp.linalg.norm(dx2)))
def test_sorted_piecewise_constant_pdf_train_mode(self):
"""Test that piecewise-constant sampling reproduces its distribution."""
batch_size = 4
num_bins = 16
num_samples = 1000000
precision = 1e5
rng = random.PRNGKey(20202020)
# Generate a series of random PDFs to sample from.
data = []
for _ in range(batch_size):
rng, key = random.split(rng)
# Randomly initialize the distances between bins.
# We're rolling our own fixed precision here to make cumsum exact.
bins_delta = jnp.round(precision * jnp.exp(
random.uniform(
key, shape=(num_bins + 1, ), minval=-3, maxval=3)))
# Set some of the bin distances to 0.
rng, key = random.split(rng)
bins_delta *= random.uniform(key, shape=bins_delta.shape) < 0.9
# Integrate the bins.
bins = jnp.cumsum(bins_delta) / precision
rng, key = random.split(rng)
bins += random.normal(key) * num_bins / 2
rng, key = random.split(rng)
# Randomly generate weights, allowing some to be zero.
weights = jnp.maximum(
0,
random.uniform(key, shape=(num_bins, ), minval=-0.5,
maxval=1.))
gt_hist = weights / weights.sum()
data.append((bins, weights, gt_hist))
# Tack on an "all zeros" weight matrix, which is a common cause of NaNs.
weights = jnp.zeros_like(weights)
gt_hist = jnp.ones_like(gt_hist) / num_bins
data.append((bins, weights, gt_hist))
bins, weights, gt_hist = [jnp.stack(x) for x in zip(*data)]
for randomized in [True, False]:
rng, key = random.split(rng)
# Draw samples from the batch of PDFs.
samples = math.sorted_piecewise_constant_pdf(
key,
bins,
weights,
num_samples,
randomized,
)
self.assertEqual(samples.shape[-1], num_samples)
# Check that samples are sorted.
self.assertTrue(jnp.all(samples[..., 1:] >= samples[..., :-1]))
# Verify that each set of samples resembles the target distribution.
for i_samples, i_bins, i_gt_hist in zip(samples, bins, gt_hist):
i_hist = jnp.float32(jnp.histogram(i_samples,
i_bins)[0]) / num_samples
i_gt_hist = jnp.array(i_gt_hist)
# Merge any of the zero-span bins until there aren't any left.
while jnp.any(i_bins[:-1] == i_bins[1:]):
j = int(jnp.where(i_bins[:-1] == i_bins[1:])[0][0])
i_hist = jnp.concatenate([
i_hist[:j],
jnp.array([i_hist[j] + i_hist[j + 1]]), i_hist[j + 2:]
])
i_gt_hist = jnp.concatenate([
i_gt_hist[:j],
jnp.array([i_gt_hist[j] + i_gt_hist[j + 1]]),
i_gt_hist[j + 2:]
])
i_bins = jnp.concatenate([i_bins[:j], i_bins[j + 1:]])
# Angle between the two histograms in degrees.
angle = 180 / jnp.pi * jnp.arccos(
jnp.minimum(
1.,
jnp.mean((i_hist * i_gt_hist) / jnp.sqrt(
jnp.mean(i_hist**2) * jnp.mean(i_gt_hist**2)))))
# Jensen-Shannon divergence.
m = (i_hist + i_gt_hist) / 2
js_div = jnp.sum(
sp.special.kl_div(i_hist, m) +
sp.special.kl_div(i_gt_hist, m)) / 2
self.assertLessEqual(angle, 0.5)
self.assertLessEqual(js_div, 1e-5)