def sdp_km(D, n_clusters):
ones = np.ones((D.shape[0], 1))
Z = cp.Semidef(D.shape[0])
objective = cp.Maximize(cp.trace(D * Z))
constraints = [Z >= 0,
Z * ones == ones,
cp.trace(Z) == n_clusters]
prob = cp.Problem(objective, constraints)
prob.solve(solver=cp.SCS, verbose=False)
Q = np.asarray(Z.value)
rs = Q.sum(axis=1)
print('Q', Q.min(), Q.max(), '|',
rs.min(), rs.max(), '|',
np.trace(Q), np.trace(D.dot(Q)))
print('Final objective', np.trace(D.dot(Q)))
return np.asarray(Z.value)
python类trace()的实例源码
def reflection_matrix(point, normal):
"""Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.
>>> v1 = numpy.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> numpy.allclose(2, numpy.trace(R))
True
>>> numpy.allclose(v0, numpy.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> numpy.allclose(v2, numpy.dot(R, v3))
True
"""
normal = unit_vector(normal[:3])
M = numpy.identity(4)
M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
return M
def reflection_matrix(point, normal):
"""Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.
>>> v1 = numpy.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> numpy.allclose(2, numpy.trace(R))
True
>>> numpy.allclose(v0, numpy.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> numpy.allclose(v2, numpy.dot(R, v3))
True
"""
normal = unit_vector(normal[:3])
M = numpy.identity(4)
M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
return M
def neg_log_likelihood(self, Lam, Theta, fixed, vary):
"compute the negative log-likelihood of the GCRF"
return -log(np.linalg.det(Lam)) + \
np.trace(np.dot(fixed.Syy, Lam) + \
2*np.dot(fixed.Sxy.T, Theta) + \
np.dot(vary.Psi, Lam))
def neg_log_likelihood_wrt_Lam(self, Lam, fixed, vary):
# compute the negative log-likelihood of the GCRF when Theta is fixed
return -log(np.linalg.det(Lam)) + \
np.trace(np.dot(fixed.Syy, Lam) + \
np.dot(vary.Psi, Lam))
def check_descent(self, newton_lambda, alpha, fixed, vary):
# check if we have made suffcient descent
DLam = np.trace(np.dot(self.grad_wrt_Lam(fixed, vary), newton_lambda)) + \
self.lamL * np.linalg.norm(self.Lam + newton_lambda, ord=1) - \
self.lamL * np.linalg.norm(self.Lam, ord=1)
nll_a = self.l1_neg_log_likelihood_wrt_Lam(self.Lam + alpha * newton_lambda, fixed, vary)
nll_b = self.l1_neg_log_likelihood_wrt_Lam(self.Lam, fixed, vary) + alpha * self.slack * DLam
return nll_a <= nll_b
def check_descent2(self, newton_lambda, alpha, fixed, vary):
lhs = self.l1_neg_log_likelihood(self.Lam + alpha*newton_lambda, self.Theta, fixed, vary)
mu = np.trace(np.dot(self.grad_wrt_Lam(fixed, vary), newton_lambda)) + \
self.lamL*self.l1_norm_off_diag(self.Lam + newton_lambda) +\
self.lamT*np.linalg.norm(self.Theta, ord=1)
rhs = self.neg_log_likelihood(self.Lam, self.Theta, fixed, vary) +\
alpha * self.slack * mu
return lhs <= rhs
def updateTransform(self):
muX = np.divide(np.sum(np.dot(self.P, self.X), axis=0), self.Np)
muY = np.divide(np.sum(np.dot(np.transpose(self.P), self.Y), axis=0), self.Np)
self.XX = self.X - np.tile(muX, (self.N, 1))
YY = self.Y - np.tile(muY, (self.M, 1))
self.A = np.dot(np.transpose(self.XX), np.transpose(self.P))
self.A = np.dot(self.A, YY)
U, _, V = np.linalg.svd(self.A, full_matrices=True)
C = np.ones((self.D, ))
C[self.D-1] = np.linalg.det(np.dot(U, V))
self.R = np.dot(np.dot(U, np.diag(C)), V)
self.YPY = np.dot(np.transpose(self.P1), np.sum(np.multiply(YY, YY), axis=1))
self.s = np.trace(np.dot(np.transpose(self.A), self.R)) / self.YPY
self.t = np.transpose(muX) - self.s * np.dot(self.R, np.transpose(muY))
def updateVariance(self):
qprev = self.q
trAR = np.trace(np.dot(self.A, np.transpose(self.R)))
xPx = np.dot(np.transpose(self.Pt1), np.sum(np.multiply(self.XX, self.XX), axis =1))
self.q = (xPx - 2 * self.s * trAR + self.s * self.s * self.YPY) / (2 * self.sigma2) + self.D * self.Np/2 * np.log(self.sigma2)
self.err = np.abs(self.q - qprev)
self.sigma2 = (xPx - self.s * trAR) / (self.Np * self.D)
if self.sigma2 <= 0:
self.sigma2 = self.tolerance / 10
def updateVariance(self):
qprev = self.q
trAB = np.trace(np.dot(self.A, np.transpose(self.B)))
xPx = np.dot(np.transpose(self.Pt1), np.sum(np.multiply(self.XX, self.XX), axis =1))
trBYPYP = np.trace(np.dot(np.dot(self.B, self.YPY), np.transpose(self.B)))
self.q = (xPx - 2 * trAB + trBYPYP) / (2 * self.sigma2) + self.D * self.Np/2 * np.log(self.sigma2)
self.err = np.abs(self.q - qprev)
self.sigma2 = (xPx - trAB) / (self.Np * self.D)
if self.sigma2 <= 0:
self.sigma2 = self.tolerance / 10
def identity_matrix():
"""Return 4x4 identity/unit matrix.
>>> I = identity_matrix()
>>> numpy.allclose(I, numpy.dot(I, I))
True
>>> numpy.sum(I), numpy.trace(I)
(4.0, 4.0)
>>> numpy.allclose(I, numpy.identity(4, dtype=numpy.float64))
True
"""
return numpy.identity(4, dtype=numpy.float64)
def reflection_matrix(point, normal):
"""Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.0
>>> v1 = numpy.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> numpy.allclose(2., numpy.trace(R))
True
>>> numpy.allclose(v0, numpy.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> numpy.allclose(v2, numpy.dot(R, v3))
True
"""
normal = unit_vector(normal[:3])
M = numpy.identity(4)
M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
return M
def rotation_from_matrix(matrix):
"""Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> angle, direc, point = rotation_from_matrix(R0)
>>> R1 = rotation_matrix(angle, direc, point)
>>> is_same_transform(R0, R1)
True
"""
R = numpy.array(matrix, dtype=numpy.float64, copy=False)
R33 = R[:3, :3]
# direction: unit eigenvector of R33 corresponding to eigenvalue of 1
l, W = numpy.linalg.eig(R33.T)
i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
direction = numpy.real(W[:, i[-1]]).squeeze()
# point: unit eigenvector of R33 corresponding to eigenvalue of 1
l, Q = numpy.linalg.eig(R)
i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(Q[:, i[-1]]).squeeze()
point /= point[3]
# rotation angle depending on direction
cosa = (numpy.trace(R33) - 1.0) / 2.0
if abs(direction[2]) > 1e-8:
sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
elif abs(direction[1]) > 1e-8:
sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
else:
sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
angle = math.atan2(sina, cosa)
return angle, direction, point
def scale_from_matrix(matrix):
"""Return scaling factor, origin and direction from scaling matrix.
>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> S0 = scale_matrix(factor, origin)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True
>>> S0 = scale_matrix(factor, origin, direct)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
M33 = M[:3, :3]
factor = numpy.trace(M33) - 2.0
try:
# direction: unit eigenvector corresponding to eigenvalue factor
l, V = numpy.linalg.eig(M33)
i = numpy.where(abs(numpy.real(l) - factor) < 1e-8)[0][0]
direction = numpy.real(V[:, i]).squeeze()
direction /= vector_norm(direction)
except IndexError:
# uniform scaling
factor = (factor + 2.0) / 3.0
direction = None
# origin: any eigenvector corresponding to eigenvalue 1
l, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no eigenvector corresponding to eigenvalue 1")
origin = numpy.real(V[:, i[-1]]).squeeze()
origin /= origin[3]
return factor, origin, direction
transformations.py 文件源码
项目：Neural-Networks-for-Inverse-Kinematics
作者: paramrajpura
项目源码
文件源码
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def identity_matrix():
"""Return 4x4 identity/unit matrix.
>>> I = identity_matrix()
>>> numpy.allclose(I, numpy.dot(I, I))
True
>>> numpy.sum(I), numpy.trace(I)
(4.0, 4.0)
>>> numpy.allclose(I, numpy.identity(4))
True
"""
return numpy.identity(4)
transformations.py 文件源码
项目：Neural-Networks-for-Inverse-Kinematics
作者: paramrajpura
项目源码
文件源码
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def rotation_from_matrix(matrix):
"""Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> angle, direc, point = rotation_from_matrix(R0)
>>> R1 = rotation_matrix(angle, direc, point)
>>> is_same_transform(R0, R1)
True
"""
R = numpy.array(matrix, dtype=numpy.float64, copy=False)
R33 = R[:3, :3]
# direction: unit eigenvector of R33 corresponding to eigenvalue of 1
w, W = numpy.linalg.eig(R33.T)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
direction = numpy.real(W[:, i[-1]]).squeeze()
# point: unit eigenvector of R33 corresponding to eigenvalue of 1
w, Q = numpy.linalg.eig(R)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(Q[:, i[-1]]).squeeze()
point /= point[3]
# rotation angle depending on direction
cosa = (numpy.trace(R33) - 1.0) / 2.0
if abs(direction[2]) > 1e-8:
sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
elif abs(direction[1]) > 1e-8:
sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
else:
sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
angle = math.atan2(sina, cosa)
return angle, direction, point
transformations.py 文件源码
项目：Neural-Networks-for-Inverse-Kinematics
作者: paramrajpura
项目源码
文件源码
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def scale_from_matrix(matrix):
"""Return scaling factor, origin and direction from scaling matrix.
>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> S0 = scale_matrix(factor, origin)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True
>>> S0 = scale_matrix(factor, origin, direct)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
M33 = M[:3, :3]
factor = numpy.trace(M33) - 2.0
try:
# direction: unit eigenvector corresponding to eigenvalue factor
w, V = numpy.linalg.eig(M33)
i = numpy.where(abs(numpy.real(w) - factor) < 1e-8)[0][0]
direction = numpy.real(V[:, i]).squeeze()
direction /= vector_norm(direction)
except IndexError:
# uniform scaling
factor = (factor + 2.0) / 3.0
direction = None
# origin: any eigenvector corresponding to eigenvalue 1
w, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no eigenvector corresponding to eigenvalue 1")
origin = numpy.real(V[:, i[-1]]).squeeze()
origin /= origin[3]
return factor, origin, direction
def rotation_from_matrix(matrix):
"""Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> angle, direc, point = rotation_from_matrix(R0)
>>> R1 = rotation_matrix(angle, direc, point)
>>> is_same_transform(R0, R1)
True
"""
R = numpy.array(matrix, dtype=numpy.float64, copy=False)
R33 = R[:3, :3]
# direction: unit eigenvector of R33 corresponding to eigenvalue of 1
w, W = numpy.linalg.eig(R33.T)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
direction = numpy.real(W[:, i[-1]]).squeeze()
# point: unit eigenvector of R33 corresponding to eigenvalue of 1
w, Q = numpy.linalg.eig(R)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(Q[:, i[-1]]).squeeze()
point /= point[3]
# rotation angle depending on direction
cosa = (numpy.trace(R33) - 1.0) / 2.0
if abs(direction[2]) > 1e-8:
sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
elif abs(direction[1]) > 1e-8:
sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
else:
sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
angle = math.atan2(sina, cosa)
return angle, direction, point
# Function to translate handshape coding to degrees of rotation, adduction, flexion
def rotation_from_matrix(matrix):
"""Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> angle, direc, point = rotation_from_matrix(R0)
>>> R1 = rotation_matrix(angle, direc, point)
>>> is_same_transform(R0, R1)
True
"""
R = numpy.array(matrix, dtype=numpy.float64, copy=False)
R33 = R[:3, :3]
# direction: unit eigenvector of R33 corresponding to eigenvalue of 1
w, W = numpy.linalg.eig(R33.T)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
direction = numpy.real(W[:, i[-1]]).squeeze()
# point: unit eigenvector of R33 corresponding to eigenvalue of 1
w, Q = numpy.linalg.eig(R)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(Q[:, i[-1]]).squeeze()
point /= point[3]
# rotation angle depending on direction
cosa = (numpy.trace(R33) - 1.0) / 2.0
if abs(direction[2]) > 1e-8:
sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
elif abs(direction[1]) > 1e-8:
sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
else:
sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
angle = math.atan2(sina, cosa)
return angle, direction, point
# Function to translate handshape coding to degrees of rotation, adduction, flexion
def identity_matrix():
"""Return 4x4 identity/unit matrix.
>>> I = identity_matrix()
>>> numpy.allclose(I, numpy.dot(I, I))
True
>>> numpy.sum(I), numpy.trace(I)
(4.0, 4.0)
>>> numpy.allclose(I, numpy.identity(4, dtype=numpy.float64))
True
"""
return numpy.identity(4, dtype=numpy.float64)
