def test_cgls_skinny(self):
tol = 1e5
A = self.A
x0 = da.ones(A.shape[1], chunks=(A.chunks[1], ))
b = atoms2.dot(A, x0)
tol_cgls = tol * da.linalg.norm(b) / da.linalg.norm(x0)
x, res, iters = cg.cgls(A, b, 0., tol=tol_cgls)
r = atoms2.dot(A, x) - b
assert da.linalg.norm(r).compute() < tol * (1 + x.size**0.5)
def test_cgls_fat(self):
tol = 1e5
A = self.B
x0 = da.ones(A.shape[1], chunks=(A.chunks[1], ))
b = atoms2.dot(A, x0)
tol_cgls = tol * da.linalg.norm(b) / da.linalg.norm(x0)
x_ln, res, iters = cg.cgls(A, b, 0., tol=tol_cgls)
r = atoms2.dot(A, x_ln) - b
assert da.linalg.norm(r).compute() < tol * (1 + x0.size**0.5)
assert (x0 - x_ln).dot(x_ln).compute() < tol * (1 + x0.size**0.5)
def cg_project(A, x, y, tol=1e-8, **options):
r""" Project (x, y) onto graph G = {(y, x) | y = Ax} via CG
In particular, form outputs as:
:math:`x_{out} = (1 + A^TA)^{-1}(A^Ty + x)`
:math:`y_{out} = Ax_{out}`
"""
fmt = 'array {} compatible'
assert A.shape[0] == y.shape[0] and A.shape[1] == x.shape[0], fmt.format(
'dims')
assert A.chunks[0] == y.chunks[0] and A.chunks[1] == x.chunks[
0], fmt.format('chunks')
token = options.pop(
'name', 'cg-project-' + dask.base.tokenize(A, x, y, tol, **options))
nm_b, nm_x, nm_y = map(lambda nm: nm + '-' + token, ('b', 'x', 'y'))
# b = A'y + x
b = atoms2.gemv(1, A, y, 1, x, transpose=True, name=nm_b)
A_hat = linop.DLORegularizedGram(A, transpose=False)
x_out, res, iters = cg.cg_graph(A_hat, b, tol=tol, name=nm_x, **options)
y_out = atoms2.dot(A, x_out, name=nm_y)
x_out, y_out = dask.persist(x_out, y_out)
return x_out, y_out, res, iters
def cgls_project(A, x, y, tol=1e-8, **options):
r""" Project (x, y) onto graph G = {(y, x) | y = Ax} via CGLS
In particular, form outputs as:
:math:`x_{out} = x + argmin_x 1/2 \|Ax' - (y - Ax)\| + 1/2 \|x'\|_2^2`
:math:`y_{out} = Ax_{out}`
"""
fmt = 'array {} compatible'
assert A.shape[0] == y.shape[0] and A.shape[1] == x.shape[0], fmt.format(
'dims')
assert A.chunks[0] == y.chunks[0] and A.chunks[1] == x.chunks[
0], fmt.format('chunks')
token = options.pop(
'name', 'cgls-project-' + dask.base.tokenize(A, x, y, tol, **options))
nm_b, nm_x, nm_y = map(lambda nm: nm + '-' + token, ('b', 'x', 'y'))
# b = y - Ax
# x_cg = argmin \|Ax' - (b)\| + \|x'\|_2^2
b = atoms2.gemv(-1, A, x, 1, y, name=nm_b)
x_cg, res, iters = cg.cgls(A, b, 1, tol=tol)
x_out = da.add(x, x_cg, name=nm_x)
y_out = atoms2.dot(A, x_out, name=nm_y)
return x_out, y_out, res, iters
def cg_reduces_residuals(A):
options = dict(graph_iters=1, maxiter=1000)
tol = 1e-5
b = da.ones(A.shape[0], chunks=A.chunks[0])
x, _, _ = cg.cg_graph(A, b)
nrm = da.linalg.norm(b - atoms2.dot(A, x)).compute()
nrm_tol = tol * (1 + da.linalg.norm(b)).compute()
return nrm < nrm_tol
def cgls(A, b, rho, **options):
b_hat = atoms2.dot(A, b, transpose=True)
A_hat = linop.DLORegularizedGram(A, regularization=rho, transpose=False)
x, _, iters = cg_graph(A_hat, b_hat, **options)
res = da.linalg.norm(atoms2.dot(A, x) - b).compute()
return x, res, iters
def dot_matches_expected(matrixlike, vec, expected, transpose=False):
tol = 1e-15 * (1 + expected.size**0.5)
output = atoms2.dot(matrixlike, vec, transpose=transpose)
norm = da.linalg.norm(output - expected).compute()
assert norm < tol, '|Ax - Ax_expected| < tol: {:.2e} < {:.2e}'.format(norm, tol)
return True