def inexact_alm_lrr_l21(X, A, lamb, display=False):
tol = 1e-8
maxIter = 1e6
d, n = X.shape
m = A.shape[1]
rho = 1.1
max_mu = 1e10
mu = 1e-6
atx = A.T.dot(X)
inv_a = np.linalg.inv(A.T.dot(A) + np.eye(m))
# Initializing optimization variables
# intialize
J = np.zeros((m, n))
Z = np.zeros((m, n))
E = np.zeros((d, n)) # sparse
Y1 = np.zeros((d, n))
Y2 = np.zeros((m, n))
# Start main loop
iter = 0
if display:
print "initial,rank=%f" % np.linalg.matrix_rank(Z)
while iter < maxIter:
iter += 1
# update J
temp = Z + Y2 / mu
U, sigma, V = np.linalg.svd(temp, 'econ')
V = V.T
svp = len(np.flatnonzero(sigma > 1.0 / mu))
if svp >= 1:
sigma = sigma[0:svp] - 1.0 / mu
else:
svp = 1
sigma = np.array([0])
J = U[:, 0:svp].dot(np.diag(sigma).dot(V[:, 0:svp].T))
# udpate Z
Z = inv_a.dot(atx - A.T.dot(E) + J + (A.T.dot(Y1) - Y2) / mu)
# update E
xmaz = X - A.dot(Z)
temp = xmaz + Y1 / mu
E = solve_l1l2(temp, lamb / mu)
leq1 = xmaz - E
leq2 = Z - J
stopC = max(np.max(np.abs(leq1)), np.max(np.abs(leq2)))
if display and (iter == 1 or np.mod(iter, 50) == 0 or stopC < tol):
print "iter", iter, ",mu=", mu, ",rank=", \
np.linalg.matrix_rank(Z, tol=1e-3*np.linalg.norm(Z, 2)), \
",stopALM=", stopC
if stopC < tol:
break
else:
Y1 += mu * leq1
Y2 += mu * leq2
mu = min(max_mu, mu * rho)
return (Z, E)
def exact_alm_lrr_l21v2(D, A, lamb, tol=1e-7, maxIter=1000, display=False):
m, n = D.shape
k = A.shape[1]
maxIter_primal = 10000
# initialize
Y = np.sign(D)
norm_two = np.linalg.norm(Y, 2)
norm_inf = np.linalg.norm(Y.flatten(1), np.inf) / lamb
dual_norm = max(norm_two, norm_inf)
Y /= dual_norm
W = np.zeros((k, n))
Z_hat = np.zeros((k, n))
E_hat = np.zeros((m, n))
# parameters
dnorm = np.linalg.norm(D, "fro")
tolProj1 = 1e-6 * dnorm
anorm = np.linalg.norm(A, 2)
tolProj2 = 1e-6 * dnorm / anorm
mu = 0.5 / norm_two # this one can be tuned
rho = 6 # this one can be tuned
# pre-computation
if m >= k:
inv_ata = np.linalg.inv(np.eye(k) + A.T.dot(A))
else:
inv_ata = np.eye(k) - np.linalg.solve((np.eye(m) + A.dot(A.T)).T, A).T.dot(A)
iter = 0
while iter < maxIter:
iter += 1
# solve the primal problem by alternative projection
primal_iter = 0
while primal_iter < maxIter_primal:
primal_iter += 1
temp_Z, temp_E = Z_hat, E_hat
# update J
temp = temp_Z + W / mu
U, S, V = np.linalg.svd(temp, "econ")
V = V.T
diagS = S
svp = len(np.flatnonzero(diagS > 1.0 / mu))
diagS = np.maximum(0, diagS - 1.0 / mu)
if svp < 0.5: # svp = 0
svp = 1
J_hat = U[:, 0:svp].dot(np.diag(diagS[0:svp]).dot(V[:, 0:svp].T))
# update Z
temp = J_hat + A.T.dot(D - temp_E) + (A.T.dot(Y) - W) / mu
Z_hat = inv_ata.dot(temp)
# update E
temp = D - A.dot(Z_hat) + Y / mu
E_hat = solve_l1l2(temp, lamb / mu)
if np.linalg.norm(E_hat - temp_E, "fro") < tolProj1 and np.linalg.norm(Z_hat - temp_Z) < tolProj2:
break
H1 = D - A.dot(Z_hat) - E_hat
H2 = Z_hat - J_hat
Y = Y + mu * H1
W = W + mu * H2
mu = rho * mu
# stop Criterion
stopCriterion = max(np.linalg.norm(H1, "fro") / dnorm, np.linalg.norm(H2, "fro") / dnorm * anorm)
if display:
print "LRR: Iteration", iter, "(", primal_iter, "), mu ", mu, ", |E|_2,0 ", np.sum(
np.sum(E_hat ** 2, 1) > 0
), ", stopCriterion ", stopCriterion
if stopCriterion < tol:
break
return (Z_hat, E_hat)
def exact_alm_lrr_l21v2(D, A, lamb, tol=1e-7, maxIter=1000, display=False):
m, n = D.shape
k = A.shape[1]
maxIter_primal = 10000
# initialize
Y = np.sign(D)
norm_two = np.linalg.norm(Y, 2)
norm_inf = np.linalg.norm(Y.flatten(1), np.inf) / lamb
dual_norm = max(norm_two, norm_inf)
Y /= dual_norm
W = np.zeros((k, n))
Z_hat = np.zeros((k, n))
E_hat = np.zeros((m, n))
# parameters
dnorm = np.linalg.norm(D, 'fro')
tolProj1 = 1e-6 * dnorm
anorm = np.linalg.norm(A, 2)
tolProj2 = 1e-6 * dnorm / anorm
mu = 0.5 / norm_two # this one can be tuned
rho = 6 # this one can be tuned
# pre-computation
if m >= k:
inv_ata = np.linalg.inv(np.eye(k) + A.T.dot(A))
else:
inv_ata = np.eye(k) - np.linalg.solve((np.eye(m) + A.dot(A.T)).T,
A).T.dot(A)
iter = 0
while iter < maxIter:
iter += 1
# solve the primal problem by alternative projection
primal_iter = 0
while primal_iter < maxIter_primal:
primal_iter += 1
temp_Z, temp_E = Z_hat, E_hat
# update J
temp = temp_Z + W / mu
U, S, V = np.linalg.svd(temp, 'econ')
V = V.T
diagS = S
svp = len(np.flatnonzero(diagS > 1.0 / mu))
diagS = np.maximum(0, diagS - 1.0 / mu)
if svp < 0.5: # svp = 0
svp = 1
J_hat = U[:, 0:svp].dot(np.diag(diagS[0:svp]).dot(V[:, 0:svp].T))
# update Z
temp = J_hat + A.T.dot(D - temp_E) + (A.T.dot(Y) - W) / mu
Z_hat = inv_ata.dot(temp)
# update E
temp = D - A.dot(Z_hat) + Y / mu
E_hat = solve_l1l2(temp, lamb / mu)
if np.linalg.norm(E_hat - temp_E, 'fro') < tolProj1 and \
np.linalg.norm(Z_hat - temp_Z) < tolProj2:
break
H1 = D - A.dot(Z_hat) - E_hat
H2 = Z_hat - J_hat
Y = Y + mu * H1
W = W + mu * H2
mu = rho * mu
# stop Criterion
stopCriterion = max(np.linalg.norm(H1, 'fro') / dnorm,
np.linalg.norm(H2, 'fro') / dnorm * anorm)
if display:
print('LRR: Iteration', iter, '(', primal_iter, '), mu ', mu, \
', |E|_2,0 ', np.sum(np.sum(E_hat ** 2, 1) > 0), \
', stopCriterion ', stopCriterion)
if stopCriterion < tol:
break
return (Z_hat, E_hat)
