Universal Regular Conditional Distributions
This provides a theoretical foundation for modeling conditional distributions in machine learning, though it appears incremental as it builds on existing attention mechanisms and approximation theory.
The paper tackles the problem of approximating regular conditional distributions (RCDs) by introducing a deep learning model called the probabilistic transformer (PT), which can universally approximate continuous functions from Euclidean space to the 1-Wasserstein space uniformly on compact sets, with strategies to avoid the curse of dimensionality.
We introduce a deep learning model that can universally approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first, it linearizes inputs from a given metric space $\mathcal{X}$ to $\mathbb{R}^d$ via a feature map, then a deep feedforward neural network processes these linearized features, and then the network's outputs are then transformed to the $1$-Wasserstein space $\mathcal{P}_1(\mathbb{R}^D)$ via a probabilistic extension of the attention mechanism of Bahdanau et al.\ (2014). Our model, called the \textit{probabilistic transformer (PT)}, 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 in which the PT avoids the curse of dimensionality when approximating $\mathcal{P}_1(\mathbb{R}^D)$-valued functions. The first strategy builds functions in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ which can be efficiently approximated by a PT, uniformly on any given compact subset of $\mathbb{R}^d$. In the second approach, given any function $f$ in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$, we build compact subsets of $\mathbb{R}^d$ whereon $f$ can be efficiently approximated by a PT.