Low Rank Matrix Completion Example

This is an example of low rank matrix completion by pure ADMM. More
explanation can be found in the code comments.

ADMM Matrix Completion/lrmcADMM.m

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+function out = lrmcADMM(Y,I,P,opts)
+
+
+%Created on 31/12/2019 by Tarmizi Adam (Email: [email protected]).
+% This code solves the noisy matrix completion problem with ADMM. The original
+% problem is the following:
+%
+% minimize_X F(X) = 0.5*||Y_omega - X_omega||^2_2 + lambda||X||_*
+%
+% We re-write the problem as the follows(used in this code)
+%
+% minimize_X F(X) = 0.5*||M_omega||^2_2 + lambda||X||_*
+% subject to, M_omega + X_omega = Y_omega,
+%
+% and solve using ADMM.
+%
+% Input: Y: The noisy and corrupted matrix (observation)
+% I: Original matrix
+% P: The mask, location of zero pixels/entries.
+% opts: Options, see "lrmcADMM_Demo" file.
+%
+% Output: Xk: Estimated low rank matrix.
+
+
+lam = opts.lam; % Regularization parameter
+Nit = opts.Nit; % Number of iteration for algorithm termination
+tol = opts.tol; % Tolerance for algorithm stopping criteria.
+
+relError = zeros(Nit,1);
+
+Xk = zeros(size(I)); % Initialize Xk
+mu = zeros(size(I)); % Lagrange multiplier
+M = Xk; % Initialize M (see objective function being solved)
+
+beta = 0.5; %Parameter related to the augmented Lagrange term.
+
+
+%% ***** Main Loop *****
+
+for k =1:Nit
+
+ X_old = Xk;
+
+ %***** X Subproblem ****
+
+ V = Xk - (P.*M + P.*Xk - Y - mu/beta);
+
+ Xk = svt(V,lam/beta);
+
+ %***** M subproblem *****
+ RHS = beta*Y - beta*(P.*Xk) + mu;
+
+ M = RHS./(beta + 1);
+
+ %***** Update Lagrange multipliers *****
+
+ mu = mu - beta*(P.*M - Y + P.*Xk);
+
+ Err = Xk - X_old;
+ relError(k) = norm(Err,'fro')/norm(Xk,'fro');
+
+ if relError(k) < tol
+ break;
+ end 
+
+end
+
+%% ***** End Main Loop *****
+
+out.sol = Xk;
+out.err = relError(1:k);
+
+
+end
+
+function Z = shrink(X,r) %Shrinkage operator
+ Z = sign(X).*max(abs(X)- r,0);
+end
+
+function Z = svt(X, r) %Singular value thresholding
+
+ [U, S, V] = svd(X);
+ s = shrink(S,r);
+
+ Z = U*s*V';
+end

ADMM Matrix Completion/lrmcADMM_Demo.m

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+% This demo files solve the low rank matrix completion problem applied to
+% image inpainting. In this demo, we apply the Alternating Direction Method
+% of Multipliers (ADMM) to do so.
+
+% Created by Tarmizi Adam on 31/12/2019. For furter assistance on the ADMM
+% algorithm used for the matrix completion problem, refer to the file
+% "lrmcADMM.m" file.
+%
+
+clc;
+clear all;
+close all;
+
+ %% For image inpainting %%
+
+ X = imread('peppers.bmp');
+ X = double(X);
+
+ [n1,n2] =size(X);
+
+%% Create projection matrix %%
+MisLvl = 0.2; % percentage of missing pixels/entries, change here.
+
+J = randperm(n1*n2);
+J = J(1:round(MisLvl*n1*n2)); 
+P = ones(n1*n2,1);
+P(J) = 0;
+P = reshape(P,[n1,n2]); % our projection matrix
+
+%% Simulate our corrupted original matrix %%
+Y = X(:);
+sigma = 30; %noise level
+noise = sigma*randn(n1*n2,1);
+
+Y = Y + noise;
+Y = reshape(Y,[n1,n2]);
+Y = P.*Y; % Our final noisy + missing entry matrix (Observation)
+
+opts.lam = 800; % Regularization parameter. Play with this and see effects.
+opts.Nit = 500; % Number of iteration for algorithm termination
+opts.tol = 1e-3;
+
+out = lrmcADMM(Y,X,P,opts); % Call ADMM solver for matrix completion.
+
+figure;
+imshow(out.sol,[]);
+
+figure; 
+imshow(X,[]);
+
+figure;
+imshow(Y,[]);
+
+
+
+
+
+
+
+