diff pyCSalgos/GAP/gap.py @ 17:ef63b89b375a

Started working on GAP, but not complete

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/pyCSalgos/GAP/gap.py	Sun Nov 06 20:58:11 2011 +0000
@@ -0,0 +1,312 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Oct 13 14:05:22 2011
+
+@author: ncleju
+"""
+
+#from numpy import *
+#from scipy import *
+import numpy as np
+import scipy as sp
+from scipy import linalg
+import math
+
+from numpy.random import RandomState
+rng = RandomState()
+
+
+
+def Generate_Analysis_Operator(d, p):
+  # generate random tight frame with equal column norms
+  if p == d:
+    T = rng.randn(d,d);
+    [Omega, discard] = np.qr(T);
+  else:
+    Omega = rng.randn(p, d);
+    T = np.zeros((p, d));
+    tol = 1e-8;
+    max_j = 200;
+    j = 1;
+    while (sum(sum(abs(T-Omega))) > np.dot(tol,np.dot(p,d)) and j < max_j):
+        j = j + 1;
+        T = Omega;
+        [U, S, Vh] = sp.linalg.svd(Omega);
+        V = Vh.T
+        #Omega = U * [eye(d); zeros(p-d,d)] * V';
+        Omega2 = np.dot(np.dot(U, np.concatenate((np.eye(d), np.zeros((p-d,d))))), V.transpose())
+        #Omega = diag(1./sqrt(diag(Omega*Omega')))*Omega;
+        Omega = np.dot(np.diag(1 / np.sqrt(np.diag(np.dot(Omega2,Omega2.transpose())))), Omega2)
+    #end
+    ##disp(j);
+#end
+  return Omega
+
+
+def Generate_Data_Known_Omega(Omega, d,p,m,k,noiselevel, numvectors, normstr):
+  #function [x0,y,M,LambdaMat] = Generate_Data_Known_Omega(Omega, d,p,m,k,noiselevel, numvectors, normstr)
+  
+  # Building an analysis problem, which includes the ingredients: 
+  #   - Omega - the analysis operator of size p*d
+  #   - M is anunderdetermined measurement matrix of size m*d (m<d)
+  #   - x0 is a vector of length d that satisfies ||Omega*x0||=p-k
+  #   - Lambda is the true location of these k zeros in Omega*x0
+  #   - a measurement vector y0=Mx0 is computed
+  #   - noise contaminated measurement vector y is obtained by
+  #     y = y0 + n where n is an additive gaussian noise with norm(n,2)/norm(y0,2) = noiselevel
+  # Added by Nic:
+  #   - Omega = analysis operator
+  #   - normstr: if 'l0', generate l0 sparse vector (unchanged). If 'l1',
+  #   generate a vector of Laplacian random variables (gamma) and
+  #   pseudoinvert to find x
+
+  # Omega is known as input parameter
+  #Omega=Generate_Analysis_Operator(d, p);
+  # Omega = randn(p,d);
+  # for i = 1:size(Omega,1)
+  #     Omega(i,:) = Omega(i,:) / norm(Omega(i,:));
+  # end
+  
+  #Init
+  LambdaMat = np.zeros((k,numvectors))
+  x0 = np.zeros((d,numvectors))
+  y = np.zeros((m,numvectors))
+  
+  M = rng.randn(m,d);
+  
+  #for i=1:numvectors
+  for i in range(0,numvectors):
+    
+    # Generate signals
+    #if strcmp(normstr,'l0')
+    if normstr == 'l0':
+        # Unchanged
+        
+        #Lambda=randperm(p); 
+        Lambda = rng.permutation(int(p));
+        Lambda = np.sort(Lambda[0:k]); 
+        LambdaMat[:,i] = Lambda; # store for output
+        
+        # The signal is drawn at random from the null-space defined by the rows 
+        # of the matreix Omega(Lambda,:)
+        [U,D,Vh] = sp.linalg.svd(Omega[Lambda,:]);
+        V = Vh.T
+        NullSpace = V[:,k:];
+        #print np.dot(NullSpace, rng.randn(d-k,1)).shape
+        #print x0[:,i].shape
+        x0[:,i] = np.squeeze(np.dot(NullSpace, rng.randn(d-k,1)));
+        # Nic: add orthogonality noise
+        #     orthonoiseSNRdb = 6;
+        #     n = randn(p,1);
+        #     #x0(:,i) = x0(:,i) + n / norm(n)^2 * norm(x0(:,i))^2 / 10^(orthonoiseSNRdb/10);
+        #     n = n / norm(n)^2 * norm(Omega * x0(:,i))^2 / 10^(orthonoiseSNRdb/10);
+        #     x0(:,i) = pinv(Omega) * (Omega * x0(:,i) + n);
+        
+    #elseif strcmp(normstr, 'l1')
+    elif normstr == 'l1':
+        print('Nic says: not implemented yet')
+        raise Exception('Nic says: not implemented yet')
+        #gamma = laprnd(p,1,0,1);
+        #x0(:,i) = Omega \ gamma;
+    else:
+        #error('normstr must be l0 or l1!');
+        print('Nic says: not implemented yet')
+        raise Exception('Nic says: not implemented yet')
+    #end
+    
+    # Acquire measurements
+    y[:,i]  = np.dot(M, x0[:,i])
+
+    # Add noise
+    t_norm = np.linalg.norm(y[:,i],2);
+    n = np.squeeze(rng.randn(m, 1));
+    y[:,i] = y[:,i] + noiselevel * t_norm * n / np.linalg.norm(n, 2);
+  #end
+
+  return x0,y,M,LambdaMat
+
+#####################
+
+#function [xhat, arepr, lagmult] = ArgminOperL2Constrained(y, M, MH, Omega, OmegaH, Lambdahat, xinit, ilagmult, params)
+def ArgminOperL2Constrained(y, M, MH, Omega, OmegaH, Lambdahat, xinit, ilagmult, params):
+  
+    #
+    # This function aims to compute
+    #    xhat = argmin || Omega(Lambdahat, :) * x ||_2   subject to  || y - M*x ||_2 <= epsilon.
+    # arepr is the analysis representation corresponding to Lambdahat, i.e.,
+    #    arepr = Omega(Lambdahat, :) * xhat.
+    # The function also returns the lagrange multiplier in the process used to compute xhat.
+    #
+    # Inputs:
+    #    y : observation/measurements of an unknown vector x0. It is equal to M*x0 + noise.
+    #    M : Measurement matrix
+    #    MH : M', the conjugate transpose of M
+    #    Omega : analysis operator
+        #    OmegaH : Omega', the conjugate transpose of Omega. Also, synthesis operator.
+    #    Lambdahat : an index set indicating some rows of Omega.
+    #    xinit : initial estimate that will be used for the conjugate gradient algorithm.
+    #    ilagmult : initial lagrange multiplier to be used in
+    #    params : parameters
+    #        params.noise_level : this corresponds to epsilon above.
+    #        params.max_inner_iteration : `maximum' number of iterations in conjugate gradient method.
+    #        params.l2_accurary : the l2 accuracy parameter used in conjugate gradient method
+    #        params.l2solver : if the value is 'pseudoinverse', then direct matrix computation (not conjugate gradient method) is used. Otherwise, conjugate gradient method is used.
+    #
+    
+    #d = length(xinit)
+    d = xinit.size
+    lagmultmax = 1e5;
+    lagmultmin = 1e-4;
+    lagmultfactor = 2;
+    accuracy_adjustment_exponent = 4/5;
+    lagmult = max(min(ilagmult, lagmultmax), lagmultmin);
+    was_infeasible = 0;
+    was_feasible = 0;
+    
+    #######################################################################
+    ## Computation done using direct matrix computation from matlab. (no conjugate gradient method.)
+    #######################################################################
+    #if strcmp(params.l2solver, 'pseudoinverse')
+    if params['solver'] == 'pseudoinverse':
+    #if strcmp(class(M), 'double') && strcmp(class(Omega), 'double')
+      if M.dtype == 'float64' and Omega.dtype == 'double':
+        while 1:
+            alpha = math.sqrt(lagmult);
+            #xhat = np.concatenate((M, alpha*Omega(Lambdahat,:)]\[y; zeros(length(Lambdahat), 1)];
+            xhat = np.concatenate((M, np.linalg.lstsq(alpha*Omega[Lambdahat,:],np.concatenate((y, np.zeros(Lambdahat.size, 1))))));
+            temp = np.linalg.norm(y - np.dot(M,xhat), 2);
+            #disp(['fidelity error=', num2str(temp), ' lagmult=', num2str(lagmult)]);
+            if temp <= params['noise_level']:
+                was_feasible = 1;
+                if was_infeasible == 1:
+                    break;
+                else:
+                    lagmult = lagmult*lagmultfactor;
+            elif temp > params['noise_level']:
+                was_infeasible = 1;
+                if was_feasible == 1:
+                    xhat = xprev;
+                    break;
+                lagmult = lagmult/lagmultfactor;
+            if lagmult < lagmultmin or lagmult > lagmultmax:
+                break;
+            xprev = xhat;
+        arepr = np.dot(Omega[Lambdahat, :], xhat);
+        return xhat,arepr,lagmult;
+
+
+    ########################################################################
+    ## Computation using conjugate gradient method.
+    ########################################################################
+    #if strcmp(class(MH),'function_handle') 
+    if hasattr(MH, '__call__'):
+        b = MH(y);
+    else:
+        b = np.dot(MH, y);
+    
+    norm_b = np.linalg.norm(b, 2);
+    xhat = xinit;
+    xprev = xinit;
+    residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
+    direction = -residual;
+    iter = 0;
+    
+    while iter < params.max_inner_iteration:
+        iter = iter + 1;
+        alpha = np.linalg.norm(residual,2)**2 / np.dot(direction.T, TheHermitianMatrix(direction, M, MH, Omega, OmegaH, Lambdahat, lagmult));
+        xhat = xhat + alpha*direction;
+        prev_residual = residual;
+        residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
+        beta = np.linalg.norm(residual,2)**2 / np.linalg.norm(prev_residual,2)**2;
+        direction = -residual + beta*direction;
+        
+        if np.linalg.norm(residual,2)/norm_b < params['l2_accuracy']*(lagmult**(accuracy_adjustment_exponent)) or iter == params['max_inner_iteration']:
+            #if strcmp(class(M), 'function_handle')
+            if hasattr(M, '__call__'):
+                temp = np.linalg.norm(y-M(xhat), 2);
+            else:
+                temp = np.linalg.norm(y-np.dot(M,xhat), 2);
+            
+            #if strcmp(class(Omega), 'function_handle')
+            if hasattr(Omega, '__call__'):
+                u = Omega(xhat);
+                u = math.sqrt(lagmult)*np.linalg.norm(u(Lambdahat), 2);
+            else:
+                u = math.sqrt(lagmult)*np.linalg.norm(Omega[Lambdahat,:]*xhat, 2);
+            
+            
+            #disp(['residual=', num2str(norm(residual,2)), ' norm_b=', num2str(norm_b), ' omegapart=', num2str(u), ' fidelity error=', num2str(temp), ' lagmult=', num2str(lagmult), ' iter=', num2str(iter)]);
+            
+            if temp <= params['noise_level']:
+                was_feasible = 1;
+                if was_infeasible == 1:
+                    break;
+                else:
+                    lagmult = lagmultfactor*lagmult;
+                    residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
+                    direction = -residual;
+                    iter = 0;
+            elif temp > params['noise_level']:
+                lagmult = lagmult/lagmultfactor;
+                if was_feasible == 1:
+                    xhat = xprev;
+                    break;
+                was_infeasible = 1;
+                residual = TheHermitianMatrix(xhat, M, MH, Omega, OmegaH, Lambdahat, lagmult) - b;
+                direction = -residual;
+                iter = 0;
+            if lagmult > lagmultmax or lagmult < lagmultmin:
+                break;
+            xprev = xhat;
+        #elseif norm(xprev-xhat)/norm(xhat) < 1e-2
+        #    disp(['rel_change=', num2str(norm(xprev-xhat)/norm(xhat))]);
+        #    if strcmp(class(M), 'function_handle')
+        #        temp = norm(y-M(xhat), 2);
+        #    else
+        #        temp = norm(y-M*xhat, 2);
+        #    end
+    #
+    #        if temp > 1.2*params.noise_level
+    #            was_infeasible = 1;
+    #            lagmult = lagmult/lagmultfactor;
+    #            xprev = xhat;
+    #        end
+    
+    #disp(['fidelity_error=', num2str(temp)]);
+    print 'fidelity_error=',temp
+    #if iter == params['max_inner_iteration']:
+        #disp('max_inner_iteration reached. l2_accuracy not achieved.');
+    
+    ##
+    # Compute analysis representation for xhat
+    ##
+    #if strcmp(class(Omega),'function_handle') 
+    if hasattr(Omega, '__call__'):
+        temp = Omega(xhat);
+        arepr = temp(Lambdahat);
+    else:    ## here Omega is assumed to be a matrix
+        arepr = np.dot(Omega[Lambdahat, :], xhat);
+    
+    return xhat,arepr,lagmult
+
+
+##
+# This function computes (M'*M + lm*Omega(L,:)'*Omega(L,:)) * x.
+##
+#function w = TheHermitianMatrix(x, M, MH, Omega, OmegaH, L, lm)
+def TheHermitianMatrix(x, M, MH, Omega, OmegaH, L, lm):
+    #if strcmp(class(M), 'function_handle')
+    if hasattr(M, '__call__'):
+        w = MH(M(x));
+    else:    ## M and MH are matrices
+        w = np.dot(np.dot(MH, M), x);
+    
+    if hasattr(Omega, '__call__'):
+        v = Omega(x);
+        vt = np.zeros(v.size);
+        vt[L] = v[L].copy();
+        w = w + lm*OmegaH(vt);
+    else:    ## Omega is assumed to be a matrix and OmegaH is its conjugate transpose
+        w = w + lm*np.dot(np.dot(OmegaH[:, L],Omega[L, :]),x);
+    
+    return w