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Universal Regular Conditional Distributions
Anastasis Kratsios
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 𝒳 to ℝᵈ 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 𝒫₁(ℝᴰ) 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 ℝᵈ to 𝒫₁(ℝᴰ) uniformly on compact sets, quantitatively. We identify two ways in which the PT avoids the curse of dimensionality when approximating 𝒫₁(ℝᴰ)-valued functions. The first strategy builds functions in C(ℝᵈ,𝒫₁(ℝᴰ)) which can be efficiently approximated by a PT, uniformly on any given compact subset of ℝᵈ. In the second approach, given any function f in C(ℝᵈ,𝒫₁(ℝᴰ)), we build compact subsets of ℝᵈ whereon f can be efficiently approximated by a PT.
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