M$^2$VAE - Derivation of a Multi-Modal Variational Autoencoder Objective from the Marginal Joint Log-Likelihood

This work gives an in-depth derivation of the trainable evidence lower bound (ELBO) obtained from the marginal joint log-Likelihood with the goal of training a multi-modal variational Autoencoder (M²VAE). I. INTRODUCTION Variational auto encoder (VAE) combine neural networks with variational inference to allow unsupervised learning of complicated distributions according to the graphical model shown in Figure 1 (left). The specific objective of VAEs is the maximization of the marginal distribution p(a) p (a z)p(z) da. II This approach proposed by [1] is used in settings where only a single modality a is present in order to find a latent encoding z (c.f. Figure 1 (left)). This work gives an in-depth derivation of the trainable evidence lower bound (ELBO) obtained from the marginal joint log-Likelihood, that satisfies all plate models as depicted in Figure 1, we are with the goal of training a multi-modal variational Autoencoder (M²VAE).