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TVproximal.f90
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TVproximal.f90
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module TVproximal
! MODULE FOR COMPUTING PROX OPERATORS FOR TV NORM
implicit none
contains
! 1-D PROXIMAL FOR L1 TV NORM
! Given a vector x in R^d, computes:
! minimize: 1/2 ||y-x||_2^2 + L||Dy||_1
! within input 'tol' of error
subroutine onedimTV1prox(x, y, tol,L, spew,iters)
integer :: n
logical, intent(in) :: spew
double precision, intent(inout), dimension(:) :: x
double precision, intent(in), dimension(:) :: y
double precision :: L, gap, tol, t, t_k_plus_one, c1 = 0.5
double precision :: f_k, f_k_plus_one, f_k_prime
double precision, dimension(:), allocatable :: diag, offdiag, u, gradient, descent_direction, sol
logical, dimension(:), allocatable :: inactive
integer :: i, j, k, info, iters, start,gradientcalc,rHessian,lapack,linesearch, project, gaptime, reduced_dim
iters = 0
n = size(x)
allocate(inactive(n-1), diag(n-1), u(n-1), gradient(n-1), offdiag(n-2))
! compute Dy for given y
do i = 1,n-1
u(i) = y(i+1) - y(i)
end do
diag = 2.0d0
offdiag = -1.0d0
! solve DDT u = Dy
call dpttrf(n-1, diag, offdiag, info)
call dpttrs(n-1, 1, diag, offdiag, u, n-1, info)
! if all constraints already satisfied
if (all(dabs(u) .le. L)) then
! compute corresponding primal optimal
x = dual_to_primal(u, y)
gradient = one_dim_TV_gradient(x)
gap = L1_duality_gap(u,gradient,L)
if (spew) then
print*,'Initially correct'
print*,'L1 TV Proximal'
print*,'final gap:', gap,'Calculated with no iterations'
end if
deallocate(inactive, diag, gradient, u, offdiag)
return
else
! Project back onto constraints
! check which constraints are satisfied
inactive = (dabs(u) < L) !inactive identifies inactive constraints
where(.not. inactive)
u = sign(L,u)
end where
end if
x = dual_to_primal(u, y)
gradient = one_dim_TV_gradient(x)
f_k = one_dim_TV_objective(x,y)
! compute initial duality gap
gap = L1_duality_gap(u,gradient,L)
allocate(descent_direction(n-1), sol(n-1))
! Iterate until convergence
! Projected Newton loop
do while (gap > tol)
call system_clock(COUNT=start)
iters = iters + 1
!construct reduced Hessian and gradient at current u
k = 0
do i = 1, n-2
if ( (inactive(i)) .or. (u(i)*gradient(i) > 0) ) then
k = k + 1
sol(k) = gradient(i)
if (inactive(i+1)) then
offdiag(k) = -1.0d0
else
offdiag(k) = 0.0d0
end if
end if
end do
if ((inactive(n-1)) .or. (u(n-1)*gradient(n-1) > 0)) then
k = k + 1
sol(k) = gradient(n-1)
end if
diag(1:k) = 2.0d0
call system_clock(COUNT = rHessian)
! solve Hd = gradient with reduced hessian for descent_direction d
! Use structure to only factor Hessian once
call dpttrf(k, diag(1:k), offdiag(1:(k-1)), info)
call dpttrs(k, 1, diag(1:k), offdiag(1:(k-1)), sol(1:k), k, info)
! map back from reduced space
k=0
do i = 1,n-1
if ((inactive(i)) .or. (u(i)*gradient(i) > 0)) then
k = k+1
descent_direction(i) = -sol(k)
else
descent_direction(i) = 0.0
end if
end do
call system_clock(COUNT=lapack)
! begin backtracking search with quadratic interp about t = 1
! attempt t = 1 step
t = 1.0d0
f_k_prime = dot_product(descent_direction, gradient) ! gradient wrt step size
! Compute potential next step
sol = u + t*descent_direction
! project onto constraints
inactive = (dabs(sol) < L) ! indicates inactive constraints
where (.not. inactive)
sol = sign(L,sol)
end where
x = dual_to_primal(sol,y)
gradient = one_dim_TV_gradient(x)
f_k_plus_one = one_dim_TV_objective(x,y)
! if insufficient decrease
if (f_k_plus_one > f_k + c1*t*f_k_prime) then
t_k_plus_one = 0.5*(f_k_prime*t**2.0) / (f_k - f_k_plus_one + t*f_k_prime) !set to quadratic interpolation minimizer
sol = u + t_k_plus_one*descent_direction
! identify inactive constraints and project
inactive = (dabs(sol) < L) ! indicates inactive constraints
where (.not. inactive)
sol = sign(L,sol)
end where
x = dual_to_primal(sol,y)
gradient = one_dim_TV_gradient(x)
f_k_plus_one = one_dim_TV_objective(x,y)
! if interpolated step is bigger than original, or insufficient decrease
if ((t_k_plus_one > t)) then
t_k_plus_one = t_k_plus_one * 0.5
sol = u + t_k_plus_one*descent_direction
! project on to constraints
inactive = (dabs(sol) < L) ! indicates inactive constraints
where (.not. inactive)
sol = sign(L,sol)
end where
x = dual_to_primal(sol,y)
gradient = one_dim_TV_gradient(x)
f_k_plus_one = one_dim_TV_objective(x,y)
end if
else
t_k_plus_one = t
end if
call system_clock(COUNT=linesearch)
! take step of size t_k_plus_one in descent_direction
u = sol
f_k = f_k_plus_one
! compute duality gap
gap = L1_duality_gap(u,gradient,L)
call system_clock(COUNT = gaptime)
if (spew) then
print*,'iter:',iters,'gap:',gap,'step:',t_k_plus_one
end if
end do
if (spew) then
print*,'L1 TV Proximal'
print*,'Final gap:', gap,'Calculated in', iters, 'iterations'
end if
!if (iters > 0) then
! print*,'TIMES:'
! print*,'Computing gradient:', gradientcalc - start
! print*,'Computing reduced Hessian:', rhessian -gradientcalc
! print*,'Computing w/ LAPACK:', lapack - gradientcalc
! print*,'Computing linesearch:', linesearch - lapack
! print*,'Computing projection:', project - linesearch
! print*,'Computing gap:', gaptime - project
! print*,'TOTAL:', gaptime - start
!end if
deallocate(inactive, diag, offdiag, u, gradient, descent_direction, sol)
return
end subroutine
! 1-D PROXIMAL FOR L2 TV NORM
! Given a vector x in R^d, computes:
! minimize: 1/2 ||y-x||_2^2 + L||Dy||_2
! within input 'tol' of error
subroutine onedimTV2prox(x, y, tol, L, spew, iters)
double precision, dimension(:), intent(in) :: y
double precision, dimension(:), intent(out) :: x
double precision, intent(in) :: L, tol
double precision, allocatable, dimension(:) :: gradient, diag, offdiag, Dy, q, u
integer :: i, iters, n, info
double precision :: gap, alpha, normu, normq
logical :: spew
iters = 0
n = size(y,1)
allocate(diag(n-1), offdiag(n-2), gradient(n-1), Dy(n-1), u(n-1), q(n-1))
alpha = 0.0d0
gap = tol + 1.0
do i = 1,n-1
Dy(i) = y(i+1) - y(i)
end do
! compute DD^T + alpha*I
diag(:) = 2.0d0 + alpha
offdiag(:) = -1.0d0
! compute cholesky factorization
call dpttrf(n-1, diag, offdiag, info)
! Solves (DD^T + alpha*I)u = Dy
u = Dy
call dpttrs(n-1,1,diag,offdiag,u,n-1,info)
open(unit=111,file='tv2convergence.txt')
gap = tol + 1.0d0
do while (gap > tol)
iters = iters + 1
! compute DD^T + alpha*I
diag(:) = 2.0d0 + alpha
offdiag(:) = -1.0d0
! compute cholesky factorization
call dpttrf(n-1, diag, offdiag, info)
! Solves (DD^T + alpha*I)u = Dy
u = Dy
call dpttrs(n-1,1,diag,offdiag,u,n-1,info)
! Compute cholesky matrix R^T
do i = 1,n-2
diag(i) = sqrt(diag(i))
offdiag(i) = offdiag(i)*diag(i)
end do
diag(n-1) = sqrt(diag(n-1))
! Solve R^Tq = u
q(1) = u(1)/diag(1)
do i = 2,n-1
q(i) = (u(i) - offdiag(i)*q(i-1))/diag(i)
end do
normu = norm2(u)
normq = norm2(q)
alpha = max(0.0d0,alpha - (1.0d0 - normu/L)*(normu/normq)**2.0d0)
! compute DD^T + alpha*I
diag(:) = 2.0d0 + alpha
offdiag(:) = -1.0d0
! compute cholesky factorization
call dpttrf(n-1, diag, offdiag, info)
! Solves (DD^T + alpha*I)u = Dy
u = Dy
call dpttrs(n-1,1,diag,offdiag,u,n-1,info)
! check stopping condition
x = dual_to_primal(u,y)
gradient = one_dim_TV_gradient(x)
gap = L2_duality_gap(u,gradient,L)
!print*,'gap:',gap
write(111,*) gap
end do
close(111)
if (spew) then
print*,'L2 TV Proximal'
print*,'final gap:', gap,'computed in',iters,'iterations'
end if
deallocate(diag, offdiag, gradient, Dy, u, q)
end subroutine
! SUBROUTINE TO COMPUTE 2-D PROX of L1 TV NORM
! Given a (m x n) matrix Y
! solves: minimize ||X-Y||_F^2 + L||X|_{tv1}
subroutine twodimTV1prox(X,Y,tol,L,spew)
double precision, dimension(:,:), intent(in) :: Y
double precision, dimension(:,:), intent(inout) :: X
double precision :: tol, L, mean_change, t
double precision, dimension(:,:), allocatable :: P, Q, Z, last
integer :: m, n, i, fill
logical :: spew
m = size(X,1)
n = size(X,2)
allocate(P(m,n), Q(m,n), Z(m,n),last(m,n))
X = Y
P = 0
Q = 0
t = 0
mean_change = 1
do while (mean_change > tol)
last = X
do i = 1,m
call onedimTV1prox(Z(i,:), X(i,:) + P(i,:), tol, L, .false., fill)
end do
P = X + P - Z
do i = 1,n
call onedimTV1prox(X(:,i), Z(:,i) + Q(:,i), tol, L, .false., fill)
end do
Q = Z + Q - X
last = dabs(last - X)
mean_change = sum(sum(last,dim=1))/(m*n)
if (spew) then
print*,'mean change:',mean_change
end if
t = t + 1
end do
if (spew) then
print*,'Converged in', t,'steps'
end if
deallocate(P,Q,Z,last)
end subroutine
! SUBROUTINE TO COMPUTE 2-D PROX of L2 TV NORM
! Given a (m x n) matrix Y
! solves: minimize ||X-Y||_F^2 + L||X||_{tv2}
subroutine twodimTV2prox(X,Y,tol,L,spew)
double precision, dimension(:,:), intent(in) :: Y
double precision, dimension(:,:), intent(inout) :: X
double precision :: tol, L, mean_change, t
double precision, dimension(:,:), allocatable :: P, Q, Z, last
logical :: spew
integer :: m, n, i, fill
m = size(X,1)
n = size(X,2)
allocate(P(m,n), Q(m,n), Z(m,n),last(m,n))
X = Y
P = 0
Q = 0
t = 0
mean_change = 1
do while (mean_change > 10D-5)
last = X
do i = 1,m
call onedimTV2prox(Z(i,:), X(i,:) + P(i,:), tol, L, .false.,fill)
end do
P = X + P - Z
do i = 1,n
call onedimTV2prox(X(:,i), Z(:,i) + Q(:,i), tol, L, .false.,fill)
end do
Q = Z + Q - X
last = dabs(last - X)
mean_change = sum(sum(last,dim=1))/(m*n)
!if (spew) then
! print*,'mean change:',mean_change
!end if
t = t + 1
end do
if (spew) then
print*,'Converged in', t,'steps'
end if
deallocate(P,Q,Z,last)
end subroutine
! FUNCTION TO COMPUTE L2 TV DUALITY GAP
! INPUT:
! dual variable u
! gradient g
! TV norm coefficient L
! returns duality gap between current dual and primal
double precision function L2_duality_gap(u,g,L)
double precision, dimension(:) :: g, u
double precision :: L
L2_duality_gap = dabs(L*norm2(g) + dot_product(u,g))
return
end function
! FUNCTION TO COMPUTE L1 TV DUALITY GAP
! INPUT:
! dual variable u
! current gradient g
! TV norm coefficient L
! Returns duality gap between primal and dual results
double precision function L1_duality_gap(u,g,L)
double precision, dimension(:) :: g, u
double precision :: L
L1_duality_gap = L*sum(dabs(g)) + dot_product(u,g)
return
end function
! FUNCTION TO COMPUTE OBJECTIVE VALUE OF 1D DUAL
! INPUT:
! current primal variable x
! TV norm input y
double precision function one_dim_TV_objective(x,y)
double precision, dimension(:) :: x,y
one_dim_TV_objective = 0.5*(dot_product(x,x) - dot_product(y, y))
return
end function
! FUNCTION TO COMPUTE GRADIENT OF OBJECTIVE FUNCTION OF 1D DUAL
! INPUT:
! current primal variable x
function one_dim_TV_gradient(x)
double precision, dimension(:) :: x
integer ::i,n
double precision, dimension(size(x,1)-1) :: one_dim_TV_gradient
n = size(x,1)
do i = 1,n-1
one_dim_TV_gradient(i) = x(i) - x(i+1)
end do
return
end function
! FUNCTION TO COMPUTE CORRESPONDING PRIMAL x for 1D TV DUAL
! INPUT:
! current dual variable u
! TV norm input y
function dual_to_primal(u,y)
double precision, dimension(:) :: u, y
integer :: i, n
double precision, dimension(size(y,1)) :: dual_to_primal
n = size(y,1)
dual_to_primal(1) = y(1) + u(1)
do i = 2,n-1
dual_to_primal(i) = y(i) - u(i-1) + u(i)
end do
dual_to_primal(n) = y(n) - u(n-1)
return
end function
function gradient_calc(X,Y,K)
double precision, dimension(:,:) :: X, Y, K
double precision, dimension(size(X,1),size(X,2)) :: gradient_calc
gradient_calc = matmul(K,X) - Y
return
end function
! given parameters mu and sigma, computes a sample from
! normal(mu,sigma)
double precision function normal_sample(mu, sigma)
double precision :: rand1, rand2, mu, sigma
call random_number(rand1)
call random_number(rand2)
! given (0,1)
normal_sample = sigma*sqrt(-2*log(rand1))*cos(2*acos(-1.0)*rand2) + mu
return
end function
end module