Alternating Minimization Algorithm

Alternating Minimization Algorithm (AMA) for 1D signal denoising. This
algorithm is slightly different from ADMM.

ADMM 1D Denoising/AMA_1D.m

@@ -0,0 +1,79 @@
+function out = AMA_1D(f,lam,rho, Nit)
+% Created by Tarmizi Adam on 24/5/2017
+%This is a simple program for 1D Signal denoising using total variation
+% denoising. The iterative algorithm used in this program is the 
+% Alternating Minimization Algorithm (AMA) proposed by Paul Tseng in the
+% paper titled:
+% "Application of a Splitting Algorithm to Decomposition in 
+% Convex Programming and Variational Inequalities"
+
+% This solver solves the following minimization problem:
+
+% min_x lambda/2*||u - f||^2_2 + ||Du||_1
+
+% Input: 
+% f = The noisy observed signal (1D-signal)
+% lam = Regularization parameter. Controls the smoothness of the 
+% denoised signal.
+% rho = Regularization parameter related to the constraint (ADMM)
+% Nit = Number of iterations to run the ADMM iterative method.
+%
+% Output:
+% out.sol = The denoised signal
+% out.funVal = Plot of the convergence of our objective
+% function
+%
+% Remarks: This code is based on the derivation in the authors ADMM notes
+% in the subsection AMA TV denoising.
+%%
+
+f = f(:);
+u = f; %Initialize
+N = length(f);
+mu = zeros(N,1); % Lagrange multiplier
+v = mu; %Initialize the v sub-problem
+funcVal = zeros(Nit,1);
+
+[D,DT,DTD] = defDDt(N);
+
+ for k = 1:Nit
+ u_old = u;
+
+ %% u sub-problem %%
+
+ u = f + (1/lam)*DT*mu; %Main difference with ADMM. No need to solve linear system.
+
+ %% v sub-problem %%
+ x = D*u - mu/rho;
+ v = shrink(x, 1/rho);
+
+ %% Update Lagrange multiplier
+ mu = mu + (v - D*u);
+
+ r1 = u - f;
+ funcVal(k) = (lam/2)*norm(r1,'fro')^2 + sum(v(:));
+
+ end
+
+ out.sol = u;
+ out.funVal = funcVal(1:k);
+
+end
+
+function [D,DT,DTD] = defDDt(N)
+%Create a first order difference matrix D
+e = ones(N,1);
+B = spdiags([e -e], [1 0], N, N);
+B(N,1) = 1;
+
+D = B; 
+clear B;
+% Create the transpose of D
+DT = D'; %Remember that DT = -D, also called the backward difference.
+DTD = D'*D;
+end
+
+function z = shrink(x,r)
+z = sign(x).*max(abs(x)- r,0);
+end
+