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