Universal Regular Conditional Distributions

We introduce a deep learning model which can generically approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first linearizes inputs from a given metric space $\mathcal{X}$ to $\mathbb{R}^d$ via a feature map then, these linearized features are processed by a deep feedforward neural network, and the network's outputs are then translated to the $1$-Wasserstein space $\mathcal{P}_1(\mathbb{R}^D)$ via a probabilistic extension of the attention mechanism introduced by Bahdanau et al. (2014). We find that the models built using our framework can approximate any continuous function from $\mathbb{R}^d$ to $\mathcal{P}_1(\mathbb{R}^D)$ uniformly on compact sets, quantitatively. We identify two ways of avoiding the curse of dimensionality when approximating $\mathcal{P}_1(\mathbb{R}^D)$-valued functions. The first strategy describes functions in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ which can be efficiently approximated on any compact subset of $\mathbb{R}^d$. The second approach describes compact subsets of $\mathbb{R}^d$, on which any most in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ can be efficiently approximated. The results are verified experimentally.