def test_solution_reconstruction():
nodes = np.array([[0, 0], [1, 0], [1, 1], [0, 1]])
point = np.array([0.1, 0.1]) # Point to interpolate.
D1 = 6
r0 = 2 # rank sample 0
r1 = 3 # rank sample 1
r2 = 4 # rank sample 2
r3 = 3 # rank sample 2
np.random.seed(1111) # For reproducibility.
# Creating a list of solutions.
Solutions = [sol0, sol1, sol2, sol3]
manifold_projection = SVDProjection(Solutions, p="max")
interpolation = GrassmannInterpolation(interpolation_method=None,
manifold_data=manifold_projection.v,
coordinates=nodes,
distance=GeodesicDistance())
interpolated_solution = interpolation.interpolate_manifold(point=point)
# assert round(interpolated_solution.data[0, 0], 9) == -0.353239531
assert round(interpolated_solution.data[0, 0], 9) == -0.316807309
def test_log_exp_maps():
sol0 = np.array([[0.61415, 1.03029, 1.02001, 0.57327, 0.79874, 0.73274],
[0.56924, 0.91700, 0.88841, 0.53737, 0.68676, 0.67751],
[0.51514, 0.87898, 0.87779, 0.47850, 0.69085, 0.61525],
[0.63038, 1.10822, 1.12313, 0.58038, 0.89142, 0.75429],
[0.69666, 1.03114, 0.95037, 0.67211, 0.71184, 0.82522],
[0.66595, 1.03789, 0.98690, 0.63420, 0.75416, 0.79110]])
sol1 = np.array([[1.05134, 1.37652, 0.95634, 0.85630, 0.47570, 1.22488],
[0.16370, 0.63105, 0.14533, 0.81030, 0.44559, 0.43358],
[1.23478, 2.10342, 1.04698, 1.68755, 0.92792, 1.73277],
[0.90538, 1.64067, 0.62027, 1.17577, 0.63644, 1.34925],
[0.58210, 0.75795, 0.65519, 0.65712, 0.37251, 0.65740],
[0.99174, 1.59375, 0.63724, 0.89107, 0.47631, 1.36581]])
sol2 = np.array([[1.04142, 0.91670, 1.47962, 1.23350, 0.94111, 0.61858],
[1.00464, 0.65684, 1.35136, 1.11288, 0.96093, 0.42340],
[1.05567, 1.33192, 1.56286, 1.43412, 0.77044, 0.97182],
[0.89812, 0.86136, 1.20204, 1.17892, 0.83788, 0.61160],
[0.46935, 0.39371, 0.63534, 0.57856, 0.47615, 0.26407],
[1.14102, 0.80869, 1.39123, 1.33076, 0.47719, 0.68170]])
sol3 = np.array([[0.60547, 0.11492, 0.78956, 0.13796, 0.76685, 0.41661],
[0.32771, 0.11606, 0.67630, 0.15208, 0.44845, 0.34840],
[0.58959, 0.10156, 0.72623, 0.11859, 0.73671, 0.38714],
[0.36283, 0.07979, 0.52824, 0.09760, 0.46313, 0.27906],
[0.87487, 0.22452, 1.30208, 0.30189, 1.22015, 0.62918],
[0.56006, 0.16879, 1.09635, 0.20431, 0.69439, 0.60317]])
# Creating a list of matrices.
matrices = [sol0, sol1, sol2, sol3]
manifold_projection = SVDProjection(matrices, p="max")
points_tangent = GrassmannOperations.log_map(
grassmann_points=manifold_projection.u,
reference_point=manifold_projection.u[0])
assert np.round(points_tangent[0][0][0], 2) == 0.0
assert np.round(points_tangent[1][0][0], 8) == 0.0
assert np.round(points_tangent[2][0][0], 8) == 0.0
assert np.round(points_tangent[3][0][0], 8) == 0.0
manifold_points = GrassmannOperations.exp_map(
tangent_points=points_tangent,
reference_point=manifold_projection.u[0])
assert np.round(manifold_points[0].data[0][0], 5) == -0.41808
assert np.round(manifold_points[1].data[0][0], 8) == -0.4180759
assert np.round(manifold_points[2].data[0][0], 8) == -0.4180759
assert np.round(manifold_points[3].data[0][0], 8) == -0.4180759
def test_kernel():
np.random.seed(1111) # For reproducibility.
from numpy.random import RandomState
rnd = RandomState(0)
# Creating a list of solutions.
Solutions = [sol0, sol1, sol2, sol3]
manifold_projection = SVDProjection(Solutions, p="max")
kernel = ProjectionKernel()
kernel.calculate_kernel_matrix(manifold_projection.u)
assert np.round(kernel.kernel_matrix[0, 1], 8) == 6.0
def test_karcher_stochastic():
matrices = [sol0, sol1, sol2, sol3]
manifold_projection = SVDProjection(matrices, p="max")
# optimization_method = GradientDescent(acceleration=True, error_tolerance=1e-4, max_iterations=1000)
psi_mean = GrassmannOperations.karcher_mean(
grassmann_points=manifold_projection.u,
optimization_method="StochasticGradientDescent",
distance=GeodesicDistance())
phi_mean = GrassmannOperations.karcher_mean(
grassmann_points=manifold_projection.v,
optimization_method="StochasticGradientDescent",
distance=GeodesicDistance())
def test_karcher():
# Creating a list of matrices.
matrices = [sol0, sol1, sol2, sol3]
manifold_projection = SVDProjection(matrices, p="max")
# optimization_method = GradientDescent(acceleration=True, error_tolerance=1e-4, max_iterations=1000)
psi_mean = GrassmannOperations.karcher_mean(
grassmann_points=manifold_projection.u,
optimization_method="GradientDescent",
distance=GeodesicDistance())
phi_mean = GrassmannOperations.karcher_mean(
grassmann_points=manifold_projection.v,
optimization_method="GradientDescent",
distance=GeodesicDistance())
assert round(psi_mean.data[0, 0], 9) == -0.387197957
ax1.imshow(Sol0)
ax2.title.set_text('Matrix 1')
ax2.imshow(Sol1)
ax3.title.set_text('Matrix 2')
ax3.imshow(Sol2)
ax4.title.set_text('Matrix 3')
ax4.imshow(Sol3)
plt.show()
# %% md
#
# Instantiate the SvdProjection class that projects the raw data to the manifold.
# %%
manifold_projection = SVDProjection(matrices, p="max")
# %% md
#
# Compute the kernels for :math:`\Psi` and :math:`\Phi`, the left and right -singular eigenvectors, respectively, of
# singular value decomposition of each solution.
# %%
projection_kernel = ProjectionKernel()
projection_kernel.calculate_kernel_matrix(points=manifold_projection.u)
kernel_psi = projection_kernel.kernel_matrix
projection_kernel.calculate_kernel_matrix(points=manifold_projection.v)
kernel_phi = projection_kernel.kernel_matrix
# Recast the data into a matrix
data_matrix = np.concatenate(
[X.reshape(-1, 1), Y.reshape(-1, 1),
Z.reshape(-1, 1)], axis=1)
# Convert each data point into a 2d array with shape (1, 3)
data = [data2dArray.reshape(1, -1).T for data2dArray in data_matrix]
#%%
#
# Project each data point onto the Grassmann manifold using SVD. Use only 300 points from the origonal data.
# Select the work with maximum rank (number of planes) from each data point (p=sys.maxsize).
# Use the matrix of left eigenvectors to calculate the kernel (KernelComposition.LEFT).
Grassmann_projection = SVDProjection(data=data[::10], p="max")
psi = Grassmann_projection.u
#%%
#
# Plot the projected data on the Grassmann (3, 1) which is a unit sphere.
Grassmann_data = np.zeros((len(psi), 3))
for i in range(len(psi)):
Grassmann_data[i, :] = psi[i].data.reshape(1, -1)
fig = plt.figure()
plt.rcParams["figure.figsize"] = (8, 8)
plt.rcParams.update({'font.size': 18})
ax = fig.add_subplot(111, projection="3d")