1 简介
In this paper, we propose a proximal splitting methodology with a non-convex penalty function based on the heavy-tailed Cauchy distribution. We first suggest a closed-form expression for calculating the proximal operator of the Cauchy prior, which then makes it applicable in generic proximal splitting algorithms. We further derive the condition required for guaranteed convergence to the global minimum in optimisation problems involving the Cauchy based penalty function. Setting the system parameters by satisfying the proposed condition ensures convergence even though the overall cost function is non-convex, when minimisation is performed via a proximal splitting algorithm. The proposed method based on Cauchy regularisation is evaluated by solving generic signal processing examples, i.e. 1D signal denoising in the frequency domain, two image reconstruction tasks including de-blurring and denoising, and error recovery in a multiple-antenna communication system. We experimentally verify the proposed convergence conditions for various cases, and show the effectiveness of the proposed Cauchy based non-convex penalty function over state-of-the-art penalty functions such as $L_1$ and total variation ( $TV$ ) norms.
2 部分代码
%% Deblurring via Cauchy proximal splitting algorithm% y = x + n% y is the 1D noisy signal% x is the clear (noise-free) signal (object of interest)% n is the additive zero-meam Gaussian noise with SNR%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Some Important Variables% ** sizeSignal: Exponent of 2 for the size of the signal in time domain.%% ** M: The size of the signal in time domain.%% ** N: The size of the signal in frequency domain.%% ** SNRdB: Noise SNR value in decibels.%% ** Niter: Maximum number of FB iterations.%% ** x: Noise-free signal%% ** y: Noisy signal%% ** mu: FB step size%% ** gamma: Cauchy scale parameter%% ** x_hat: The reconstructed signal in frequency domain.%% ** x_Cauchy: The reconstructed signal in frequency domain.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LICENSE%% This program is free software: you can redistribute it and/or modify% it under the terms of the GNU General Public License as published by% the Free Software Foundation, either version 3 of the License, or% (at your option) any later version.%% This program is distributed in the hope that it will be useful,% but WITHOUT ANY WARRANTY; without even the implied warranty of% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the% GNU General Public License for more details.%% You should have received a copy of the GNU General Public License% along with this program. If not, see <https://www.gnu.org/licenses/>.%% Copyright (C) Oktay Karakus,PhD% University of Bristol, UK% o.karakus@bristol.ac.uk% April 2020%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCE%% [1] O Karakus, P Mayo, and A Achim. "Convergence Guarantees for% Non-Convex Optimisation with Cauchy-Based Penalties"% arXiv preprint.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clearvarsclose allclc%% Parameter InitializationsizeSignal = 7;M = 2^sizeSignal;N = 2^(sizeSignal + 2);SNRdB = 3;rmse = @(err) sqrt(mean(abs(err(:)).^2));truncate = @(x, M) x(1:M);AH = @(x) fft(x, N)/sqrt(N);A = @(X) truncate(ifft(X), M) * sqrt(N);Niter = 500;[x,y] = wnoise(3, sizeSignal, SNRdB);x = x';y = y';%% Cauchyx_hat = AH(zeros(size(y))); % regularized resultiter = 1;old_X = x_hat;grad_f_x = @(x) AH(A(x) - y); % gradient operatorxx = ones(size(y));yy = 0*ones(size(y));Lip = norm(grad_f_x(xx) - grad_f_x(yy), 2)/norm(xx - yy, 2); % A general calculation for Lipschitz constant.mu = 1.5/Lip;gamma = 2*sqrt(mu)/2;delta_x = inf;tic;while (delta_x(iter) > 1e-3) && (iter < Niter) iter = iter + 1; Z = x_hat - mu*(AH(A(x_hat) - y)); x_hat = CauchyProx(real(Z), gamma, mu); delta_x(iter) = max(abs( x_hat(:) - old_X(:) )) / max(abs(old_X(:))); % Error calculation old_X = x_hat;end x_Cauchy = A(x_hat);timeSim = toc;RMSE_noisy = rmse(x - y);RMSE_regularized = rmse(x - x_Cauchy);fprintf('Cauchy proximal splitting (CPS) for 1D denoising\nSolved after %d iterations in %.3f seconds\nNoisy RMSE = %.3f\nReconstructed RMSE = %.3f\n', iter, timeSim, RMSE_noisy, RMSE_regularized)figure;set(gcf, 'Position', [100 100 800 300])subplot('Position', [0.0501, 0.1001, 0.9, 0.85])plot(x, 'b', 'Linewidth', 1.5)hold onplot(y, 'k-.', 'Linewidth', 1)plot(x_Cauchy, 'r--', 'Linewidth', 2)grid onlegend('Noise-free', ['Noisy (SNR = ' num2str(SNRdB) ' dB)'], 'CPS')text(40, 0.9*max(y), ['RMSE_{Noisy} = ' num2str(RMSE_noisy)], 'Color', 'Black')text(40, 0.6*max(y), ['RMSE_{CPS} = ' num2str(RMSE_regularized)], 'Color', 'Red')
3 仿真结果
4 参考文献
[1] Karakus O , Mayo P , Achim A . Convergence Guarantees for Non-Convex Optimisation with Cauchy-Based Penalties[J]. 2020.
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