  Simultaneously sampling from a complex distribution with intractable
normalizing constant and approximating expectations under this distribution is
a notoriously challenging problem. We introduce a novel scheme, Invertible Flow
Non Equilibrium Sampling (InFine), which departs from classical Sequential
Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) approaches. InFine
constructs unbiased estimators of expectations and in particular of normalizing
constants by combining the orbits of a deterministic transform started from
random initializations.When this transform is chosen as an appropriate
integrator of a conformal Hamiltonian system, these orbits are optimization
paths. InFine is also naturally suited to design new MCMC sampling schemes by
selecting samples on the optimization paths.Additionally, InFine can be used to
construct an Evidence Lower Bound (ELBO) leading to a new class of Variational
AutoEncoders (VAE).
