Conditional Deep Inverse Rosenblatt Transports

by   Tiangang Cui, et al.

We present a novel offline-online method to mitigate the computational burden of the characterization of conditional beliefs in statistical learning. In the offline phase, the proposed method learns the joint law of the belief random variables and the observational random variables in the tensor-train (TT) format. In the online phase, it utilizes the resulting order-preserving conditional transport map to issue real-time characterization of the conditional beliefs given new observed information. Compared with the state-of-the-art normalizing flows techniques, the proposed method relies on function approximation and is equipped with thorough performance analysis. This also allows us to further extend the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional belief variables. On the one hand, we present novel heuristics to reorder and/or reparametrize the variables to enhance the approximation power of TT. On the other, we integrate the TT-based transport maps and the parameter reordering/reparametrization into layered compositions to further improve the performance of the resulting transport maps. We demonstrate the efficiency of the proposed method on various statistical learning tasks in ordinary differential equations (ODEs) and partial differential equations (PDEs).


page 22

page 23

page 24


Deep Composition of Tensor Trains using Squared Inverse Rosenblatt Transports

Characterising intractable high-dimensional random variables is one of t...

Multilinear POD-DEIM model reduction for 2D and 3D nonlinear systems of differential equations

We are interested in the numerical solution of coupled nonlinear partial...

High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations

In this paper we develop a new machinery to study the capacity of artifi...

Solving high-dimensional parabolic PDEs using the tensor train format

High-dimensional partial differential equations (PDEs) are ubiquitous in...

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

We develop new dynamically orthogonal tensor methods to approximate mult...

Conditional indifference and conditional preservation

The idea of preserving conditional beliefs emerged recently as a new par...