def test_beale(self):
x0 = np.zeros(2)
x, iter = gd(of.beale, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.array([3, 0.5])))
def test_bohachevsky3(self):
x0 = np.ones(2) * 200
x, iter = gd(of.bohachevsky3, of.grad_bohachevsky3, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_ackley(self):
# This has many minima, make sure we only get the closest one
x0 = np.zeros(10)
x, iter = gd(of.ackley, of.grad_ackley, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_sphere(self):
x0 = np.random.random(10)
x, iter = gd(of.sphere, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_maxiter(self):
x0 = 6
num_iter = 5
with self.assertWarnsRegex(Warning, 'GD hit maxiters!'):
x, iter = gd(of.quadratic, fd_complex_step, x0, iter=num_iter)
self.assertEqual(iter, num_iter)
def test_rastrigin(self):
x0 = np.ones(10)
x, iter = gd(of.rastrigin, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def cosamp(A,y,k,lstsq='exact',tol=1e-8,maxiter=500,x=None,disp=False):
'''Compressive sampling matching pursuit (CoSaMP) algorithm.
A -- Measurement matrix.
y -- Measurements (i.e., y = Ax).
k -- Number of expected nonzero coefficients.
lstsq -- How to solve intermediate least squares problem.
tol -- Stopping criteria.
maxiter -- Maximum number of iterations.
x -- True signal we are trying to estimate.
disp -- Whether or not to display iterations.
lstsq function:
lstsq = { 'exact', 'lm', 'gd' }.
'exact' solves it using numpy's linalg.lstsq method.
'lm' uses solves with the Levenberg-Marquardt algorithm.
'gd' uses 3 iterations of a gradient descent solver.
Implements Algorithm 8.7 from:
Eldar, Yonina C., and Gitta Kutyniok, eds. Compressed sensing: theory
and applications. Cambridge University Press, 2012.
'''
# length of measurement vector and original signal
n,N = A.shape[:]
# Initializations
x_hat = np.zeros(N,dtype=y.dtype)
r = y.copy()
# Decide how we want to solve the intermediate least squares problem
if lstsq == 'exact':
lstsq_fun = lambda A0,y: np.linalg.lstsq(A0,y,rcond=None)[0]
elif lstsq == 'lm':
from scipy.optimize import least_squares
lstsq_fun = lambda A0,y: least_squares(lambda x: np.linalg.norm(y - np.dot(A0,x)),np.zeros(A0.shape[1],dtype=y.dtype))['x']
elif lstsq == 'gd':
# This doesn't work very well...
from mr_utils.optimization import gd,fd_complex_step
lstsq_fun = lambda A0,y: gd(lambda x: np.linalg.norm(y - np.dot(A0,x)),fd_complex_step,np.zeros(A0.shape[1],dtype=y.dtype),iter=3)[0]
else:
raise NotImplementedError()
# Start up a table
if disp:
table = Table([ 'iter','residual','MSE' ],[ len(repr(maxiter)),8,8 ],[ 'd','e','e' ])
hdr = table.header()
for line in hdr.split('\n'):
logging.info(line)
for ii in range(maxiter):
# Get step direction
g = np.dot(A.T,r)
# Add 2*k largest elements of g to support set
Tn = np.union1d(x_hat.nonzero()[0],np.argsort(np.abs(g))[-(2*k):])
# Solve the least squares problem
xn = np.zeros(N)
xn[Tn] = lstsq_fun(A[:,Tn],y)
xn[np.argsort(np.abs(xn))[:-k]] = 0
x_hat = xn.copy()
# Compute new residual
r = y - np.dot(A,x_hat)
# Compute stopping criteria
stop_criteria = np.linalg.norm(r)/np.linalg.norm(y)
# Show MSE at current iteration if we wanted it
if disp:
logging.info(table.row([ ii,stop_criteria,np.mean((np.abs(x - x_hat)**2)) ]))
# Check stopping criteria
if stop_criteria < tol:
break
return(x_hat)
def test_quadratic(self):
x0 = 6
x, iter = gd(of.quadratic, fd_complex_step, x0, alpha=None)
self.assertTrue(np.allclose(x, 9 / 4))
def test_beale(self):
'''Beale function with min at (3, 1/2).'''
x0 = np.zeros(2)
x, _iter = gd(of.beale, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.array([3, 0.5])))
def test_bohachevsky3(self):
'''Bohachevsky function 3 with min at origin.'''
x0 = np.ones(2) * 200
x, _iter = gd(of.bohachevsky3, of.grad_bohachevsky3, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_sphere(self):
'''Sphere function with min at origin.'''
x0 = np.random.random(10)
x, _iter = gd(of.sphere, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_ackley(self):
'''Ackley function with min at 0.'''
# This has many minima, make sure we only get the closest one
x0 = np.zeros(10)
x, _iter = gd(of.ackley, of.grad_ackley, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_rastrigin(self):
'''Rastrigin function with min at 0.'''
x0 = np.ones(10)
x, _iter = gd(of.rastrigin, fd_complex_step, x0)
self.assertTrue(np.allclose(x, np.zeros(x.shape)))
def test_maxiter(self):
'''Make sure that hitting maxiter raises a warning.'''
x0 = 6
num_iter = 5
with self.assertWarnsRegex(Warning, 'GD hit maxiters!'):
_x, _iter = gd(of.quadratic, fd_complex_step, x0, maxiter=num_iter)
def test_quadratic(self):
'''Simple quadratic function with min at 9/4.'''
x0 = 6
x, _iter = gd(of.quadratic, fd_complex_step, x0, alpha=None)
self.assertTrue(np.allclose(x, 9 / 4))