Multimodal Unsupervised Image-to-Image Translation-Supplementary Material

Proof. Let z1 denote the latent code, which is the concatenation of c1 and s1. We denote the encoded latent distribution by pE(z1), which is defined by z1 = E1(x1) and x1 sampled from the data distribution p(x1). We denote the latent distribution at generation time by p(z1), which is obtained by s1 ∼ q(s1) and c1 ∼ p(c2). The generated image distribution pG(x1) = p(x2→1) is defined by x1 = G1(z1) and z1 sampled from p(z1). According to the change of variable formula for probability density functions: