def ADMM_sparse_RPCA(D, max_it=10):
"""
Alternating Direction Method of Multipliers for
Robust Principal Component Analysis (a.k.a. Principal Component Pursuit)
min ||X||_* + lambda*||Y||_1 s.t. X + Y = D
Sparse SVD: compute only the first 'sv' singular values, where sv is
an estimate of the number of singular values that are greater than
the value 1/rho, used in the shrinkage operation.
A different approach with sparse SVD: Lin, Z., Chen, M., & Ma, Y. (2010). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv preprint arXiv:1009.5055.
"""
(M,N) = D.shape
X = zeros((M,N))
Y = zeros((M,N))
Z = zeros((M,N))
l = 1 / sqrt(max(M,N))
print l
rho = float(M)*N/(4*sparse.one_norm(D))
print rho
k = 0
epsilon = 0.0000001 * linalg.norm(D)
residual = epsilon * 100
print epsilon
sv = 20
while k < max_it and residual >= epsilon:
# Update X
print "CSC it"
TMP = scipy.sparse.csc_matrix(D-Y-Z)
print "SVD it"
U,sigma,Vh = scipy.sparse.linalg.svds(TMP, k = sv)
#Ut,sigma,Vh = sparsesvd(TMP, k = sv)
print sv, sum(sigma >= 1/rho)
if sum(sigma >= 1/rho) < sv:
sv += 1
else:
sv = int(min(sv + (k+1)*0.05*N,N))
print "Get X"
X = dot(dot(U, diag(sparse.shrinkage(sigma,1/rho))),Vh)
# Update Y
# Element-wise shrinkage
print "Shrink it"
Y = sparse.shrinkage(D - X - Z, l/rho)
# Update Z
Z = Z + X + Y - D
print "Norm it"
residual = linalg.norm(D-X-Y)
#print k, 1/2*linalg.norm(dot(A,x_star)-s) + l*linalg.norm(x_star,1)
k += 1
print k, residual, sum(Y!=0)
return X, Y
def ADMM_RPCA(D, max_it=10):
"""
Alternating Direction Method of Multipliers for
Robust Principal Component Analysis (a.k.a. Principal Component Pursuit)
min ||X||_* + lambda*||Y||_1 s.t. X + Y = D
Original source: Candes, E. J., Li, X., Ma, Y., & Wright, J. (2009). Robust principal component analysis?. arXiv preprint arXiv:0912.3599.
An implementation: http://www.cds.caltech.edu/~ipapusha/code/pcp_admm.m
"""
(M,N) = D.shape
l = 1 / sqrt(max(M,N))
print "Lambda", l
rho = 10*float(M)*N/(4*sparse.one_norm(D))
print "Rho:", rho
k = 0
X,Y,Z = zeros((M,N)), zeros((M,N)), zeros((M,N))
epsilon = 0.0000001 * linalg.norm(D)
residual = epsilon * 100
print "Primal tolerance", epsilon
print "Max iterations", max_it
while k < max_it and residual >= epsilon:
# Update X
print "SVD it"
U,sigma,Vh = linalg.svd(D - Y - Z, full_matrices=False)
print "Get X"
X = dot(dot(U, diag(sparse.shrinkage(sigma,1/rho))),Vh)
# Update Y
# Element-wise shrinkage
print "Shrink it"
newY = sparse.shrinkage(D - X - Z, l/rho)
# Update Z
Z = Z + X + newY - D
print "Norm it"
residual = linalg.norm(D - X - Y)
dual_residual = rho*linalg.norm(newY - Y)
Y = newY
#print k, 1/2*linalg.norm(dot(A,x_star)-s) + l*linalg.norm(x_star,1)
#TODO vary rho
#TODO dual_tolerance
k += 1
print k, residual, dual_residual, sum(Y!=0)
return X, Y
