num_rows = 128
subspace_dimension = 3
matrix = rnd.randn(num_rows, num_rows)
matrix = 0.5 * (matrix + matrix.T)
cost, egrad, ehess = create_cost_egrad_ehess(
backend, matrix, subspace_dimension)
manifold = Grassmann(num_rows, subspace_dimension)
problem = pymanopt.Problem(manifold, cost=cost, egrad=egrad, ehess=ehess)
if quiet:
problem.verbosity = 0
solver = TrustRegions()
estimated_spanning_set = solver.solve(
problem, Delta_bar=8*np.sqrt(subspace_dimension))
if quiet:
return
eigenvalues, eigenvectors = la.eig(matrix)
column_indices = np.argsort(eigenvalues)[-subspace_dimension:]
spanning_set = eigenvectors[:, column_indices]
print("Geodesic distance between true and estimated dominant subspace:",
manifold.dist(spanning_set, estimated_spanning_set))
if __name__ == "__main__":
runner = ExampleRunner(run, "Dominant invariant subspace",
SUPPORTED_BACKENDS)
runner.run()
cost, egrad, ehess = create_cost_egrad_ehess(backend, samples,
num_components)
manifold = Stiefel(dimension, num_components)
problem = pymanopt.Problem(manifold, cost, egrad=egrad, ehess=ehess)
if quiet:
problem.verbosity = 0
solver = TrustRegions()
# from pymanopt.solvers import ConjugateGradient
# solver = ConjugateGradient()
estimated_span_matrix = solver.solve(problem)
if quiet:
return
estimated_projector = estimated_span_matrix @ estimated_span_matrix.T
eigenvalues, eigenvectors = np.linalg.eig(samples.T @ samples)
indices = np.argsort(eigenvalues)[::-1][:num_components]
span_matrix = eigenvectors[:, indices]
projector = span_matrix @ span_matrix.T
print(
"Frobenius norm error between estimated and closed-form projection "
"matrix:", np.linalg.norm(projector - estimated_projector))
if __name__ == "__main__":
runner = ExampleRunner(run, "PCA", SUPPORTED_BACKENDS)
runner.run()
problem.verbosity = 0
solver = ConjugateGradient()
left_singular_vectors, singular_values, right_singular_vectors = \
solver.solve(problem)
low_rank_approximation = (left_singular_vectors @ np.diag(singular_values)
@ right_singular_vectors)
if not quiet:
u, s, vt = la.svd(matrix, full_matrices=False)
indices = np.argsort(s)[-rank:]
low_rank_solution = (
u[:, indices] @ np.diag(s[indices]) @ vt[indices, :])
print("Analytic low-rank solution:")
print()
print(low_rank_solution)
print()
print("Rank-{} approximation:".format(rank))
print()
print(low_rank_approximation)
print()
print("Frobenius norm error:",
la.norm(low_rank_approximation - low_rank_solution))
print()
if __name__ == "__main__":
runner = ExampleRunner(run, "Low-rank matrix approximation",
SUPPORTED_BACKENDS)
runner.run()
(
cost,
euclidean_gradient,
euclidean_hessian,
) = create_cost_and_derivates(manifold, samples, targets, backend)
problem = pymanopt.Problem(
manifold,
cost,
euclidean_gradient=euclidean_gradient,
euclidean_hessian=euclidean_hessian,
)
estimated_weights = optimizer.run(problem).point
if not quiet:
print(f"Run {k + 1}")
print(
"Weights found by pymanopt (top) / "
"closed form solution (bottom)"
)
print(estimated_weights)
print(np.linalg.pinv(samples) @ targets)
print("")
if __name__ == "__main__":
runner = ExampleRunner(
run, "Multiple linear regression", SUPPORTED_BACKENDS
)
runner.run()
# zero, optimization first becomes much slower, than simply doesn't work
# anymore because of floating point overflow errors (NaN's and Inf's start
# to appear). If it is too large, then log-sum-exp is a poor approximation
# of the max function, and the spread will be less uniform. An okay value
# seems to be 0.01 or 0.001 for example. Note that a better strategy than
# using a small epsilon straightaway is to reduce epsilon bit by bit and to
# warm-start subsequent optimization in that way. Trustregions will be more
# appropriate for these fine tunings.
epsilon = 0.0015
cost = create_cost(backend, dimension, num_points, epsilon)
manifold = Elliptope(num_points, dimension)
problem = pymanopt.Problem(manifold, cost)
if quiet:
problem.verbosity = 0
solver = ConjugateGradient(mingradnorm=1e-8, maxiter=1e5)
Yopt = solver.solve(problem)
if quiet:
return
Xopt = Yopt @ Yopt.T
maxdot = np.triu(Xopt, 1).max()
print("Maximum angle between any two points:", maxdot)
if __name__ == "__main__":
runner = ExampleRunner(run, "Packing on the sphere", SUPPORTED_BACKENDS)
runner.run()
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
m = 5
n = 8
matrix = rnd.randn(m, n)
cost, egrad = create_cost_egrad(backend, matrix)
manifold = Oblique(m, n)
problem = pymanopt.Problem(manifold, cost=cost, egrad=egrad)
if quiet:
problem.verbosity = 0
solver = ConjugateGradient()
Xopt = solver.solve(problem)
if quiet:
return
# Calculate the actual solution by normalizing the columns of A.
X = matrix / la.norm(matrix, axis=0)[np.newaxis, :]
# Print information about the solution.
print("Solution found: %s" % np.allclose(X, Xopt, rtol=1e-3))
print("Frobenius-error: %f" % la.norm(X - Xopt))
if __name__ == "__main__":
runner = ExampleRunner(run, "Closest unit Frobenius norm approximation",
SUPPORTED_BACKENDS)
runner.run()
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
num_rows = 10
rank = 3
matrix = rnd.randn(num_rows, num_rows)
matrix = 0.5 * (matrix + matrix.T)
# Solve the problem with pymanopt.
cost, egrad, ehess = create_cost_egrad_ehess(backend, matrix, rank)
manifold = Oblique(rank, num_rows)
problem = pymanopt.Problem(manifold, cost, egrad=egrad, ehess=ehess)
if quiet:
problem.verbosity = 0
solver = TrustRegions()
X = solver.solve(problem)
if quiet:
return
C = X.T @ X
print("Diagonal elements:", np.diag(C))
print("Eigenvalues:", np.sort(la.eig(C)[0].real)[::-1])
if __name__ == "__main__":
runner = ExampleRunner(run, "Nearest low-rank correlation matrix",
SUPPORTED_BACKENDS)
runner.run()
solver = SteepestDescent()
estimated_dominant_eigenvector = solver.solve(problem)
if quiet:
return
# Calculate the actual solution by a conventional eigenvalue decomposition.
eigenvalues, eigenvectors = la.eig(matrix)
dominant_eigenvector = eigenvectors[:, np.argmax(eigenvalues)]
# Make sure both vectors have the same direction. Both are valid
# eigenvectors, but for comparison we need to get rid of the sign
# ambiguity.
if (np.sign(dominant_eigenvector[0]) != np.sign(
estimated_dominant_eigenvector[0])):
estimated_dominant_eigenvector = -estimated_dominant_eigenvector
# Print information about the solution.
print("l2-norm of x: %f" % la.norm(dominant_eigenvector))
print("l2-norm of xopt: %f" % la.norm(estimated_dominant_eigenvector))
print("Solution found: %s" % np.allclose(
dominant_eigenvector, estimated_dominant_eigenvector, rtol=1e-3))
error_norm = la.norm(dominant_eigenvector - estimated_dominant_eigenvector)
print("l2-error: %f" % error_norm)
if __name__ == "__main__":
runner = ExampleRunner(run, "Dominant eigenvector of a PSD matrix",
SUPPORTED_BACKENDS)
runner.run()
return (U * J) @ Vt
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
n = 3
m = 10
k = 10
A = np.random.randn(k, n, m)
B = np.random.randn(k, n, m)
ABt = np.array([Ak @ Bk.T for Ak, Bk in zip(A, B)])
cost, egrad = create_cost_egrad(backend, ABt)
manifold = SpecialOrthogonalGroup(n, k)
problem = pymanopt.Problem(manifold, cost, egrad=egrad)
if quiet:
problem.verbosity = 0
solver = SteepestDescent()
X = solver.solve(problem)
if not quiet:
Xopt = np.array([compute_optimal_solution(ABtk) for ABtk in ABt])
print("Frobenius norm error:", np.linalg.norm(Xopt - X))
if __name__ == "__main__":
runner = ExampleRunner(run, "Optimal rotations example",
SUPPORTED_BACKENDS)
runner.run()
from examples._tools import ExampleRunner
from pymanopt.manifolds import Positive
from pymanopt.tools.diagnostics import check_retraction
SUPPORTED_BACKENDS = ("numpy", )
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
m = 128
n = 64
manifold = Positive(m, n, k=2)
check_retraction(manifold)
if __name__ == "__main__":
runner = ExampleRunner(run, "Check retraction of positive manifold",
SUPPORTED_BACKENDS)
runner.run()
@pymanopt.function.Theano(x)
def cost(x):
return -x.T.dot(T.dot(A, x))
else:
raise ValueError("Unsupported backend '{:s}'".format(backend))
return cost, egrad
def run(backend=SUPPORTED_BACKENDS[0], quiet=True):
# Generate random problem data.
n = 128
A = np.random.randn(n, n)
A = 0.5 * (A + A.T)
cost, egrad = create_cost_egrad(backend, A)
# Create the problem structure.
manifold = Sphere(n)
problem = Problem(manifold=manifold, cost=cost, egrad=egrad)
if quiet:
problem.verbosity = 0
# Numerically check gradient consistency (optional).
check_gradient(problem)
if __name__ == "__main__":
runner = ExampleRunner(run, "Check gradient for sphere manifold",
SUPPORTED_BACKENDS)
runner.run()