10,154 research outputs found
Geodesics in the space of measure-preserving maps and plans
We study Brenier's variational models for incompressible Euler equations.
These models give rise to a relaxation of the Arnold distance in the space of
measure-preserving maps and, more generally, measure-preserving plans. We
analyze the properties of the relaxed distance, we show a close link between
the Lagrangian and the Eulerian model, and we derive necessary and sufficient
optimality conditions for minimizers. These conditions take into account a
modified Lagrangian induced by the pressure field. Moreover, adapting some
ideas of Shnirelman, we show that, even for non-deterministic final conditions,
generalized flows can be approximated in energy by flows associated to
measure-preserving maps
MixFlows: principled variational inference via mixed flows
This work presents mixed variational flows (MixFlows), a new variational
family that consists of a mixture of repeated applications of a map to an
initial reference distribution. First, we provide efficient algorithms for
i.i.d. sampling, density evaluation, and unbiased ELBO estimation. We then show
that MixFlows have MCMC-like convergence guarantees when the flow map is
ergodic and measure-preserving, and provide bounds on the accumulation of error
for practical implementations where the flow map is approximated. Finally, we
develop an implementation of MixFlows based on uncorrected discretized
Hamiltonian dynamics combined with deterministic momentum refreshment.
Simulated and real data experiments show that MixFlows can provide more
reliable posterior approximations than several black-box normalizing flows, as
well as samples of comparable quality to those obtained from state-of-the-art
MCMC methods
Mixed Variational Flows for Discrete Variables
Variational flows allow practitioners to learn complex continuous
distributions, but approximating discrete distributions remains a challenge.
Current methodologies typically embed the discrete target in a continuous space
- usually via continuous relaxation or dequantization - and then apply a
continuous flow. These approaches involve a surrogate target that may not
capture the original discrete target, might have biased or unstable gradients,
and can create a difficult optimization problem. In this work, we develop a
variational flow family for discrete distributions without any continuous
embedding. First, we develop a measure-preserving and discrete (MAD) invertible
map that leaves the discrete target invariant, and then create a mixed
variational flow (MAD Mix) based on that map. Our family provides access to
i.i.d. sampling and density evaluation with virtually no tuning effort. We also
develop an extension to MAD Mix that handles joint discrete and continuous
models. Our experiments suggest that MAD Mix produces more reliable
approximations than continuous-embedding flows while being significantly faster
to train
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
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