huber_1d_demo
Ivan Selesnick
June 2016
Contents
Define Huber function
huber = @(t) min( 0.5 * t.^2, min( abs(t-1) + 0.5, abs(t+1) + 0.5 ));
Huber function
T = 3;
t = linspace(-T, T, 101);
figure(1)
clf
plot(t, huber(t));
xlabel('x')
title('Huber function')
axis([-T T 0 T])
axis image
drawnow
print -dpdf -bestfit figures/Huber_1d
MC penalty
figure(2)
clf
plot(t, abs(t) - huber(t), [-T 0 T], [T 0 T], '--')
xlabel('x')
title('MC penalty function');
axis([-T T 0 T])
axis image
txt1 = text(1.5, 1.7, '|x|');
txt2 = text(1.5, 0.6, '\phi(x)');
drawnow
print -dpdf -bestfit figures/MC_penalty_1d
scaled functions
sb = @(t, b) huber(b^2 * t) / b^2;
b_list = [1 0.5 sqrt(2) 1/sqrt(2)];
figure(1)
clf
subplot(2, 1, 1)
for b = b_list
line(t, sb(t, b));
text(T+0.1, T-0.5/b^2, sprintf('b = %.2f', b));
end
xlabel('x')
axis image
axis([-T T 0 T])
title('Scaled Huber function s_b(x)');
subplot(2, 1, 2)
for b = b_list
line(t, abs(t) - sb(t, b));
text(T+0.1, 0.5/b^2, sprintf('b = %.2f', b));
end
line([-T 0 T], [T 0 T], 'linestyle', '--')
xlabel('x')
axis([-T T 0 T])
axis image
title('Scaled MC penalty \phi_b(x)');
drawnow
print -dpdf -bestfit figures/scaled_funs_1d
