nikcleju@17: # -*- coding: utf-8 -*-
nikcleju@17: """
nikcleju@17: Created on Thu Oct 13 14:05:22 2011
nikcleju@17: 
nikcleju@17: @author: ncleju
nikcleju@17: """
nikcleju@17: 
nikcleju@17: #from numpy import *
nikcleju@17: #from scipy import *
nikcleju@17: import numpy as np
nikcleju@17: import scipy as sp
nikcleju@17: from scipy import linalg
nikcleju@17: import math
nikcleju@17: 
nikcleju@17: from numpy.random import RandomState
nikcleju@17: rng = RandomState()
nikcleju@17: 
nikcleju@17: 
nikcleju@17: 
nikcleju@17: def Generate_Analysis_Operator(d, p):
nikcleju@17:   # generate random tight frame with equal column norms
nikcleju@17:   if p == d:
nikcleju@17:     T = rng.randn(d,d);
nikcleju@17:     [Omega, discard] = np.qr(T);
nikcleju@17:   else:
nikcleju@17:     Omega = rng.randn(p, d);
nikcleju@17:     T = np.zeros((p, d));
nikcleju@17:     tol = 1e-8;
nikcleju@17:     max_j = 200;
nikcleju@17:     j = 1;
nikcleju@17:     while (sum(sum(abs(T-Omega))) > np.dot(tol,np.dot(p,d)) and j < max_j):
nikcleju@17:         j = j + 1;
nikcleju@17:         T = Omega;
nikcleju@17:         [U, S, Vh] = sp.linalg.svd(Omega);
nikcleju@17:         V = Vh.T
nikcleju@17:         #Omega = U * [eye(d); zeros(p-d,d)] * V';
nikcleju@17:         Omega2 = np.dot(np.dot(U, np.concatenate((np.eye(d), np.zeros((p-d,d))))), V.transpose())
nikcleju@17:         #Omega = diag(1./sqrt(diag(Omega*Omega')))*Omega;
nikcleju@17:         Omega = np.dot(np.diag(1 / np.sqrt(np.diag(np.dot(Omega2,Omega2.transpose())))), Omega2)
nikcleju@17:     #end
nikcleju@17:     ##disp(j);
nikcleju@17: #end
nikcleju@17:   return Omega
nikcleju@17: 
nikcleju@17: 
nikcleju@17: def Generate_Data_Known_Omega(Omega, d,p,m,k,noiselevel, numvectors, normstr):
nikcleju@17:   #function [x0,y,M,LambdaMat] = Generate_Data_Known_Omega(Omega, d,p,m,k,noiselevel, numvectors, normstr)
nikcleju@17:   
nikcleju@17:   # Building an analysis problem, which includes the ingredients: 
nikcleju@17:   #   - Omega - the analysis operator of size p*d
nikcleju@17:   #   - M is anunderdetermined measurement matrix of size m*d (m<d)
nikcleju@17:   #   - x0 is a vector of length d that satisfies ||Omega*x0||=p-k
nikcleju@17:   #   - Lambda is the true location of these k zeros in Omega*x0
nikcleju@17:   #   - a measurement vector y0=Mx0 is computed
nikcleju@17:   #   - noise contaminated measurement vector y is obtained by
nikcleju@17:   #     y = y0 + n where n is an additive gaussian noise with norm(n,2)/norm(y0,2) = noiselevel
nikcleju@17:   # Added by Nic:
nikcleju@17:   #   - Omega = analysis operator
nikcleju@17:   #   - normstr: if 'l0', generate l0 sparse vector (unchanged). If 'l1',
nikcleju@17:   #   generate a vector of Laplacian random variables (gamma) and
nikcleju@17:   #   pseudoinvert to find x
nikcleju@17: 
nikcleju@17:   # Omega is known as input parameter
nikcleju@17:   #Omega=Generate_Analysis_Operator(d, p);
nikcleju@17:   # Omega = randn(p,d);
nikcleju@17:   # for i = 1:size(Omega,1)
nikcleju@17:   #     Omega(i,:) = Omega(i,:) / norm(Omega(i,:));
nikcleju@17:   # end
nikcleju@17:   
nikcleju@17:   #Init
nikcleju@17:   LambdaMat = np.zeros((k,numvectors))
nikcleju@17:   x0 = np.zeros((d,numvectors))
nikcleju@17:   y = np.zeros((m,numvectors))
nikcleju@17:   
nikcleju@17:   M = rng.randn(m,d);
nikcleju@17:   
nikcleju@17:   #for i=1:numvectors
nikcleju@17:   for i in range(0,numvectors):
nikcleju@17:     
nikcleju@17:     # Generate signals
nikcleju@17:     #if strcmp(normstr,'l0')
nikcleju@17:     if normstr == 'l0':
nikcleju@17:         # Unchanged
nikcleju@17:         
nikcleju@17:         #Lambda=randperm(p); 
nikcleju@17:         Lambda = rng.permutation(int(p));
nikcleju@17:         Lambda = np.sort(Lambda[0:k]); 
nikcleju@17:         LambdaMat[:,i] = Lambda; # store for output
nikcleju@17:         
nikcleju@17:         # The signal is drawn at random from the null-space defined by the rows 
nikcleju@17:         # of the matreix Omega(Lambda,:)
nikcleju@17:         [U,D,Vh] = sp.linalg.svd(Omega[Lambda,:]);
nikcleju@17:         V = Vh.T
nikcleju@17:         NullSpace = V[:,k:];
nikcleju@17:         #print np.dot(NullSpace, rng.randn(d-k,1)).shape
nikcleju@17:         #print x0[:,i].shape
nikcleju@17:         x0[:,i] = np.squeeze(np.dot(NullSpace, rng.randn(d-k,1)));
nikcleju@17:         # Nic: add orthogonality noise
nikcleju@17:         #     orthonoiseSNRdb = 6;
nikcleju@17:         #     n = randn(p,1);
nikcleju@17:         #     #x0(:,i) = x0(:,i) + n / norm(n)^2 * norm(x0(:,i))^2 / 10^(orthonoiseSNRdb/10);
nikcleju@17:         #     n = n / norm(n)^2 * norm(Omega * x0(:,i))^2 / 10^(orthonoiseSNRdb/10);
nikcleju@17:         #     x0(:,i) = pinv(Omega) * (Omega * x0(:,i) + n);
nikcleju@17:         
nikcleju@17:     #elseif strcmp(normstr, 'l1')
nikcleju@17:     elif normstr == 'l1':
nikcleju@17:         print('Nic says: not implemented yet')
nikcleju@17:         raise Exception('Nic says: not implemented yet')
nikcleju@17:         #gamma = laprnd(p,1,0,1);
nikcleju@17:         #x0(:,i) = Omega \ gamma;
nikcleju@17:     else:
nikcleju@17:         #error('normstr must be l0 or l1!');
nikcleju@17:         print('Nic says: not implemented yet')
nikcleju@17:         raise Exception('Nic says: not implemented yet')
nikcleju@17:     #end
nikcleju@17:     
nikcleju@17:     # Acquire measurements
nikcleju@17:     y[:,i]  = np.dot(M, x0[:,i])
nikcleju@17: 
nikcleju@17:     # Add noise
nikcleju@17:     t_norm = np.linalg.norm(y[:,i],2);
nikcleju@17:     n = np.squeeze(rng.randn(m, 1));
nikcleju@17:     y[:,i] = y[:,i] + noiselevel * t_norm * n / np.linalg.norm(n, 2);
nikcleju@17:   #end
nikcleju@17: 
nikcleju@17:   return x0,y,M,LambdaMat
nikcleju@17: 
nikcleju@17: #####################
nikcleju@17: 
nikcleju@17: #function [xhat, arepr, lagmult] = ArgminOperL2Constrained(y, M, MH, Omega, OmegaH, Lambdahat, xinit, ilagmult, params)
nikcleju@17: def ArgminOperL2Constrained(y, M, MH, Omega, OmegaH, Lambdahat, xinit, ilagmult, params):
nikcleju@17:   
nikcleju@17:     #
nikcleju@17:     # This function aims to compute
nikcleju@17:     #    xhat = argmin || Omega(Lambdahat, :) * x ||_2   subject to  || y - M*x ||_2 <= epsilon.
nikcleju@17:     # arepr is the analysis representation corresponding to Lambdahat, i.e.,
nikcleju@17:     #    arepr = Omega(Lambdahat, :) * xhat.
nikcleju@17:     # The function also returns the lagrange multiplier in the process used to compute xhat.
nikcleju@17:     #
nikcleju@17:     # Inputs:
nikcleju@17:     #    y : observation/measurements of an unknown vector x0. It is equal to M*x0 + noise.
nikcleju@17:     #    M : Measurement matrix
nikcleju@17:     #    MH : M', the conjugate transpose of M
nikcleju@17:     #    Omega : analysis operator
nikcleju@17:         #    OmegaH : Omega', the conjugate transpose of Omega. Also, synthesis operator.
nikcleju@17:     #    Lambdahat : an index set indicating some rows of Omega.
nikcleju@17:     #    xinit : initial estimate that will be used for the conjugate gradient algorithm.
nikcleju@17:     #    ilagmult : initial lagrange multiplier to be used in
nikcleju@17:     #    params : parameters
nikcleju@17:     #        params.noise_level : this corresponds to epsilon above.
nikcleju@17:     #        params.max_inner_iteration : `maximum' number of iterations in conjugate gradient method.
nikcleju@17:     #        params.l2_accurary : the l2 accuracy parameter used in conjugate gradient method
nikcleju@17:     #        params.l2solver : if the value is 'pseudoinverse', then direct matrix computation (not conjugate gradient method) is used. Otherwise, conjugate gradient method is used.
nikcleju@17:     #
nikcleju@17:     
nikcleju@17:     #d = length(xinit)
nikcleju@17:     d = xinit.size
nikcleju@17:     lagmultmax = 1e5;
nikcleju@17:     lagmultmin = 1e-4;
nikcleju@17:     lagmultfactor = 2;
nikcleju@17:     accuracy_adjustment_exponent = 4/5;
nikcleju@17:     lagmult = max(min(ilagmult, lagmultmax), lagmultmin);
nikcleju@17:     was_infeasible = 0;
nikcleju@17:     was_feasible = 0;
nikcleju@17:     
nikcleju@17:     #######################################################################
nikcleju@17:     ## Computation done using direct matrix computation from matlab. (no conjugate gradient method.)
nikcleju@17:     #######################################################################
nikcleju@17:     #if strcmp(params.l2solver, 'pseudoinverse')
nikcleju@17:     if params['solver'] == 'pseudoinverse':
nikcleju@17:     #if strcmp(class(M), 'double') && strcmp(class(Omega), 'double')
nikcleju@17:       if M.dtype == 'float64' and Omega.dtype == 'double':
nikcleju@17:         while 1:
nikcleju@17:             alpha = math.sqrt(lagmult);
nikcleju@17:             #xhat = np.concatenate((M, alpha*Omega(Lambdahat,:)]\[y; zeros(length(Lambdahat), 1)];
nikcleju@17:             xhat = np.concatenate((M, np.linalg.lstsq(alpha*Omega[Lambdahat,:],np.concatenate((y, np.zeros(Lambdahat.size, 1))))));
nikcleju@17:             temp = np.linalg.norm(y - np.dot(M,xhat), 2);
nikcleju@17:             #disp(['fidelity error=', num2str(temp), ' lagmult=', num2str(lagmult)]);
nikcleju@17:             if temp <= params['noise_level']:
nikcleju@17:                 was_feasible = 1;
nikcleju@17:                 if was_infeasible == 1:
nikcleju@17:                     break;
nikcleju@17:                 else:
nikcleju@17:                     lagmult = lagmult*lagmultfactor;
nikcleju@17:             elif temp > params['noise_level']:
nikcleju@17:                 was_infeasible = 1;
nikcleju@17:                 if was_feasible == 1:
nikcleju@17:                     xhat = xprev;
nikcleju@17:                     break;
nikcleju@17:                 lagmult = lagmult/lagmultfactor;
nikcleju@17:             if lagmult < lagmultmin or lagmult > lagmultmax:
nikcleju@17:                 break;
nikcleju@17:             xprev = xhat;
nikcleju@17:         arepr = np.dot(Omega[Lambdahat, :], xhat);
nikcleju@17:         return xhat,arepr,lagmult;
nikcleju@17: 
nikcleju@17: 
nikcleju@17:     ########################################################################
nikcleju@17:     ## Computation using conjugate gradient method.
nikcleju@17:     ########################################################################
nikcleju@17:     #if strcmp(class(MH),'function_handle') 
nikcleju@17:     if hasattr(MH, '__call__'):
nikcleju@17:         b = MH(y);
nikcleju@17:     else:
nikcleju@17:         b = np.dot(MH, y);
nikcleju@17:     
nikcleju@17:     norm_b = np.linalg.norm(b, 2);
nikcleju@17:     xhat = xinit;
nikcleju@17:     xprev = xinit;
nikcleju@17:     residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
nikcleju@17:     direction = -residual;
nikcleju@17:     iter = 0;
nikcleju@17:     
nikcleju@17:     while iter < params.max_inner_iteration:
nikcleju@17:         iter = iter + 1;
nikcleju@17:         alpha = np.linalg.norm(residual,2)**2 / np.dot(direction.T, TheHermitianMatrix(direction, M, MH, Omega, OmegaH, Lambdahat, lagmult));
nikcleju@17:         xhat = xhat + alpha*direction;
nikcleju@17:         prev_residual = residual;
nikcleju@17:         residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
nikcleju@17:         beta = np.linalg.norm(residual,2)**2 / np.linalg.norm(prev_residual,2)**2;
nikcleju@17:         direction = -residual + beta*direction;
nikcleju@17:         
nikcleju@17:         if np.linalg.norm(residual,2)/norm_b < params['l2_accuracy']*(lagmult**(accuracy_adjustment_exponent)) or iter == params['max_inner_iteration']:
nikcleju@17:             #if strcmp(class(M), 'function_handle')
nikcleju@17:             if hasattr(M, '__call__'):
nikcleju@17:                 temp = np.linalg.norm(y-M(xhat), 2);
nikcleju@17:             else:
nikcleju@17:                 temp = np.linalg.norm(y-np.dot(M,xhat), 2);
nikcleju@17:             
nikcleju@17:             #if strcmp(class(Omega), 'function_handle')
nikcleju@17:             if hasattr(Omega, '__call__'):
nikcleju@17:                 u = Omega(xhat);
nikcleju@17:                 u = math.sqrt(lagmult)*np.linalg.norm(u(Lambdahat), 2);
nikcleju@17:             else:
nikcleju@17:                 u = math.sqrt(lagmult)*np.linalg.norm(Omega[Lambdahat,:]*xhat, 2);
nikcleju@17:             
nikcleju@17:             
nikcleju@17:             #disp(['residual=', num2str(norm(residual,2)), ' norm_b=', num2str(norm_b), ' omegapart=', num2str(u), ' fidelity error=', num2str(temp), ' lagmult=', num2str(lagmult), ' iter=', num2str(iter)]);
nikcleju@17:             
nikcleju@17:             if temp <= params['noise_level']:
nikcleju@17:                 was_feasible = 1;
nikcleju@17:                 if was_infeasible == 1:
nikcleju@17:                     break;
nikcleju@17:                 else:
nikcleju@17:                     lagmult = lagmultfactor*lagmult;
nikcleju@17:                     residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
nikcleju@17:                     direction = -residual;
nikcleju@17:                     iter = 0;
nikcleju@17:             elif temp > params['noise_level']:
nikcleju@17:                 lagmult = lagmult/lagmultfactor;
nikcleju@17:                 if was_feasible == 1:
nikcleju@17:                     xhat = xprev;
nikcleju@17:                     break;
nikcleju@17:                 was_infeasible = 1;
nikcleju@17:                 residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
nikcleju@17:                 direction = -residual;
nikcleju@17:                 iter = 0;
nikcleju@17:             if lagmult > lagmultmax or lagmult < lagmultmin:
nikcleju@17:                 break;
nikcleju@17:             xprev = xhat;
nikcleju@17:         #elseif norm(xprev-xhat)/norm(xhat) < 1e-2
nikcleju@17:         #    disp(['rel_change=', num2str(norm(xprev-xhat)/norm(xhat))]);
nikcleju@17:         #    if strcmp(class(M), 'function_handle')
nikcleju@17:         #        temp = norm(y-M(xhat), 2);
nikcleju@17:         #    else
nikcleju@17:         #        temp = norm(y-M*xhat, 2);
nikcleju@17:         #    end
nikcleju@17:     #
nikcleju@17:     #        if temp > 1.2*params.noise_level
nikcleju@17:     #            was_infeasible = 1;
nikcleju@17:     #            lagmult = lagmult/lagmultfactor;
nikcleju@17:     #            xprev = xhat;
nikcleju@17:     #        end
nikcleju@17:     
nikcleju@17:     #disp(['fidelity_error=', num2str(temp)]);
nikcleju@17:     print 'fidelity_error=',temp
nikcleju@17:     #if iter == params['max_inner_iteration']:
nikcleju@17:         #disp('max_inner_iteration reached. l2_accuracy not achieved.');
nikcleju@17:     
nikcleju@17:     ##
nikcleju@17:     # Compute analysis representation for xhat
nikcleju@17:     ##
nikcleju@17:     #if strcmp(class(Omega),'function_handle') 
nikcleju@17:     if hasattr(Omega, '__call__'):
nikcleju@17:         temp = Omega(xhat);
nikcleju@17:         arepr = temp(Lambdahat);
nikcleju@17:     else:    ## here Omega is assumed to be a matrix
nikcleju@17:         arepr = np.dot(Omega[Lambdahat, :], xhat);
nikcleju@17:     
nikcleju@17:     return xhat,arepr,lagmult
nikcleju@17: 
nikcleju@17: 
nikcleju@17: ##
nikcleju@17: # This function computes (M'*M + lm*Omega(L,:)'*Omega(L,:)) * x.
nikcleju@17: ##
nikcleju@17: #function w = TheHermitianMatrix(x, M, MH, Omega, OmegaH, L, lm)
nikcleju@17: def TheHermitianMatrix(x, M, MH, Omega, OmegaH, L, lm):
nikcleju@17:     #if strcmp(class(M), 'function_handle')
nikcleju@17:     if hasattr(M, '__call__'):
nikcleju@17:         w = MH(M(x));
nikcleju@17:     else:    ## M and MH are matrices
nikcleju@17:         w = np.dot(np.dot(MH, M), x);
nikcleju@17:     
nikcleju@17:     if hasattr(Omega, '__call__'):
nikcleju@17:         v = Omega(x);
nikcleju@17:         vt = np.zeros(v.size);
nikcleju@17:         vt[L] = v[L].copy();
nikcleju@17:         w = w + lm*OmegaH(vt);
nikcleju@17:     else:    ## Omega is assumed to be a matrix and OmegaH is its conjugate transpose
nikcleju@17:         w = w + lm*np.dot(np.dot(OmegaH[:, L],Omega[L, :]),x);
nikcleju@17:     
nikcleju@17:     return w