prox_tnn.m

function [X,tnn,trank] = prox_tnn(Y,rho)

% The proximal operator of the tensor nuclear norm of a 3 way tensor
%
% min_X rho*||X||_*+0.5*||X-Y||_F^2
%
% Y     -    n1*n2*n3 tensor
%
% X     -    n1*n2*n3 tensor
% tnn   -    tensor nuclear norm of X
% trank -    tensor tubal rank of X
%
% version 2.1 - 14/06/2018
%
% Written by Canyi Lu (canyilu@gmail.com)
%
%
% References: 
% Canyi Lu, Tensor-Tensor Product Toolbox. Carnegie Mellon University. 
% June, 2018. https://github.com/canyilu/tproduct.
%
% Canyi Lu, Jiashi Feng, Yudong Chen, Wei Liu, Zhouchen Lin and Shuicheng
% Yan, Tensor Robust Principal Component Analysis with A New Tensor Nuclear
% Norm, arXiv preprint arXiv:1804.03728, 2018
%

[n1,n2,n3] = size(Y);
X = zeros(n1,n2,n3);
Y = fft(Y,[],3);
tnn = 0;
trank = 0;
        
% first frontal slice
[U,S,V] = svd(Y(:,:,1),'econ');
S = diag(S);
r = length(find(S>rho));
if r>=1
    S = S(1:r)-rho;
    X(:,:,1) = U(:,1:r)*diag(S)*V(:,1:r)';
    tnn = tnn+sum(S);
    trank = max(trank,r);
end
% i=2,...,halfn3
halfn3 = round(n3/2);
for i = 2 : halfn3
    [U,S,V] = svd(Y(:,:,i),'econ');
    S = diag(S);
    r = length(find(S>rho));
    if r>=1
        S = S(1:r)-rho;
        X(:,:,i) = U(:,1:r)*diag(S)*V(:,1:r)';
        tnn = tnn+sum(S)*2;
        trank = max(trank,r);
    end
    X(:,:,n3+2-i) = conj(X(:,:,i));
end

% if n3 is even
if mod(n3,2) == 0
    i = halfn3+1;
    [U,S,V] = svd(Y(:,:,i),'econ');
    S = diag(S);
    r = length(find(S>rho));
    if r>=1
        S = S(1:r)-rho;
        X(:,:,i) = U(:,1:r)*diag(S)*V(:,1:r)';
        tnn = tnn+sum(S);
        trank = max(trank,r);
    end
end
tnn = tnn/n3;
X = ifft(X,[],3);