1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
|
% tveq_newton.m
%
% Newton algorithm for log-barrier subproblems for TV minimization
% with equality constraints.
%
% Usage:
% [xp,tp,niter] = tveq_newton(x0, t0, A, At, b, tau,
% newtontol, newtonmaxiter, slqtol, slqmaxiter)
%
% x0,t0 - starting points
%
% A - Either a handle to a function that takes a N vector and returns a K
% vector , or a KxN matrix. If A is a function handle, the algorithm
% operates in "largescale" mode, solving the Newton systems via the
% Conjugate Gradients algorithm.
%
% At - Handle to a function that takes a K vector and returns an N vector.
% If A is a KxN matrix, At is ignored.
%
% b - Kx1 vector of observations.
%
% tau - Log barrier parameter.
%
% newtontol - Terminate when the Newton decrement is <= newtontol.
%
% newtonmaxiter - Maximum number of iterations.
%
% slqtol - Tolerance for SYMMLQ; ignored if A is a matrix.
%
% slqmaxiter - Maximum number of iterations for SYMMLQ; ignored
% if A is a matrix.
%
% Written by: Justin Romberg, Caltech
% Email: jrom@acm.caltech.edu
% Created: October 2005
%
function [xp, tp, niter] = tveq_newton(x0, t0, A, At, b, tau, newtontol, newtonmaxiter, slqtol, slqmaxiter)
largescale = isa(A,'function_handle');
alpha = 0.01;
beta = 0.5;
N = length(x0);
n = round(sqrt(N));
K = length(b);
% create (sparse) differencing matrices for TV
Dv = spdiags([reshape([-ones(n-1,n); zeros(1,n)],N,1) ...
reshape([zeros(1,n); ones(n-1,n)],N,1)], [0 1], N, N);
Dh = spdiags([reshape([-ones(n,n-1) zeros(n,1)],N,1) ...
reshape([zeros(n,1) ones(n,n-1)],N,1)], [0 n], N, N);
% auxillary matrices for preconditioning
Mdv = spdiags([reshape([ones(n-1,n); zeros(1,n)],N,1) ...
reshape([zeros(1,n); ones(n-1,n)],N,1)], [0 1], N, N);
Mdh = spdiags([reshape([ones(n,n-1) zeros(n,1)],N,1) ...
reshape([zeros(n,1) ones(n,n-1)],N,1)], [0 n], N, N);
Mmd = reshape([ones(n-1,n-1) zeros(n-1,1); zeros(1,n)],N,1);
% initial point
x = x0;
t = t0;
Dhx = Dh*x; Dvx = Dv*x;
ft = 1/2*(Dhx.^2 + Dvx.^2 - t.^2);
f = sum(t) - (1/tau)*(sum(log(-ft)));
niter = 0;
done = 0;
while (~done)
ntgx = Dh'*((1./ft).*Dhx) + Dv'*((1./ft).*Dvx);
ntgt = -tau - t./ft;
gradf = -(1/tau)*[ntgx; ntgt];
sig22 = 1./ft + (t.^2)./(ft.^2);
sig12 = -t./ft.^2;
sigb = 1./ft.^2 - (sig12.^2)./sig22;
w1p = ntgx - Dh'*(Dhx.*(sig12./sig22).*ntgt) - Dv'*(Dvx.*(sig12./sig22).*ntgt);
wp = [w1p; zeros(K,1)];
if (largescale)
% diagonal of H11p
dg11p = Mdh'*(-1./ft + sigb.*Dhx.^2) + Mdv'*(-1./ft + sigb.*Dvx.^2) + 2*Mmd.*sigb.*Dhx.*Dvx;
afac = max(dg11p);
hpfun = @(z) Hpeval(z, A, At, Dh, Dv, Dhx, Dvx, sigb, ft, afac);
[dxv,slqflag,slqres,slqiter] = symmlq(hpfun, wp, slqtol, slqmaxiter);
if (slqres > 1/2)
disp('Cannot solve system. Returning previous iterate. (See Section 4 of notes for more information.)');
xp = x;
return
end
else
H11p = Dh'*sparse(diag(-1./ft + sigb.*Dhx.^2))*Dh + ...
Dv'*sparse(diag(-1./ft + sigb.*Dvx.^2))*Dv + ...
Dh'*sparse(diag(sigb.*Dhx.*Dvx))*Dv + ...
Dv'*sparse(diag(sigb.*Dhx.*Dvx))*Dh;
afac = max(diag(H11p));
Hp = full([H11p afac*A'; afac*A zeros(K)]);
%keyboard
opts.SYM = true;
[dxv, hcond] = linsolve(Hp, wp, opts);
if (hcond < 1e-14)
disp('Matrix ill-conditioned. Returning previous iterate. (See Section 4 of notes for more information.)');
xp = x; tp = t;
return
end
end
dx = dxv(1:N);
Dhdx = Dh*dx; Dvdx = Dv*dx;
dt = (1./sig22).*(ntgt - sig12.*(Dhx.*Dhdx + Dvx.*Dvdx));
% minimum step size that stays in the interior
aqt = Dhdx.^2 + Dvdx.^2 - dt.^2;
bqt = 2*(Dhdx.*Dhx + Dvdx.*Dvx - t.*dt);
cqt = Dhx.^2 + Dvx.^2 - t.^2;
tsols = [(-bqt+sqrt(bqt.^2-4*aqt.*cqt))./(2*aqt); ...
(-bqt-sqrt(bqt.^2-4*aqt.*cqt))./(2*aqt) ];
indt = find([(bqt.^2 > 4*aqt.*cqt); (bqt.^2 > 4*aqt.*cqt)] & (tsols > 0));
smax = min(1, min(tsols(indt)));
s = (0.99)*smax;
% line search
suffdec = 0;
backiter = 0;
while (~suffdec)
xp = x + s*dx; tp = t + s*dt;
Dhxp = Dhx + s*Dhdx; Dvxp = Dvx + s*Dvdx;
ftp = 1/2*(Dhxp.^2 + Dvxp.^2 - tp.^2);
fp = sum(tp) - (1/tau)*(sum(log(-ftp)));
flin = f + alpha*s*(gradf'*[dx; dt]);
suffdec = (fp <= flin);
s = beta*s;
backiter = backiter + 1;
if (backiter > 32)
disp('Stuck backtracking, returning last iterate. (See Section 4 of notes for more information.)');
xp = x; tp = t;
return
end
end
% set up for next iteration
x = xp; t = tp;
Dvx = Dvxp; Dhx = Dhxp;
ft = ftp; f = fp;
lambda2 = -(gradf'*[dx; dt]);
stepsize = s*norm([dx; dt]);
niter = niter + 1;
done = (lambda2/2 < newtontol) | (niter >= newtonmaxiter);
disp(sprintf('Newton iter = %d, Functional = %8.3f, Newton decrement = %8.3f, Stepsize = %8.3e', ...
niter, f, lambda2/2, stepsize));
if (largescale)
disp(sprintf(' SYMMLQ Res = %8.3e, SYMMLQ Iter = %d', slqres, slqiter));
else
disp(sprintf(' H11p condition number = %8.3e', hcond));
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Implicit application of Hessian
function y = Hpeval(z, A, At, Dh, Dv, Dhx, Dvx, sigb, ft, afac)
N = length(ft);
K = length(z)-N;
w = z(1:N);
v = z(N+1:N+K);
Dhw = Dh*w;
Dvw = Dv*w;
y1 = Dh'*((-1./ft + sigb.*Dhx.^2).*Dhw + sigb.*Dhx.*Dvx.*Dvw) + ...
Dv'*((-1./ft + sigb.*Dvx.^2).*Dvw + sigb.*Dhx.*Dvx.*Dhw) + afac*At(v);
y2 = afac*A(w);
y = [y1; y2];
|
