Exchangeable Generative Models with Flow Scans

Abstract

In this work, we develop a new approach to generative density estimation for exchangeable, non-i.i.d. data. The proposed framework, FlowScan, combines invertible flow transformations with a sorted scan to flexibly model the data while preserving exchangeability. Unlike most existing methods, FlowScan exploits the intradependencies within sets to learn both global and local structure. FlowScan represents the first approach that is able to apply sequential methods to exchangeable density estimation without resorting to averaging over all possible permutations. We achieve new state-of-the-art performance on point cloud and image set modeling.

Targeted free energy estimation via learned mappings

“…( 22), we can easily see that the above property implies that σ leaves J M k , and hence J M , invariant, because we sum over all particles and the sum is permutation invariant. Previous studies have made similar observations 40,41 . Our implementation of C is based on the architecture proposed by Vaswani et al 42 , often referred to as the transformer, which we use in a permutation-equivariant configuration.…”

Exchangeable Neural ODE for Set Modeling

“…BRUNO [4] employs an independent flow transformation for each element and an exchangeable student-t process for the invariant likelihood. FlowScan [5] transforms the set likelihood problem to the familiar sequence likelihood problem via a scan sorting operation. In this work, we extend a flow based generative model for exchangeable sets with a tractable invariant likelihood.…”

“…We extend the continuous normalizing flow proposed in [6,15] to model exchangeable sets x ∈ X n . Specifically, we have the following proposition from [5], repeated here for convenience: Proposition 1. For a flow model with transformation q(•) and base likelihood p Z (•), the input likelihood p X (x) = p Z (q(x)) det dq dx is exchangeable iff the transformation is permutation equivariant and the base likelihood is invariant.…”

“…Set generative models with tractable, exchangeable likelihoods have recently been investigated (that is, likelihoods which are invariant to permutations) [3,4,5]. Simple approaches that estimate likelihood for each instance independently are insufficient since global structures cannot be inferred.…”

Normalizing flows for atomic solids

“…As also shown by Refs. [24,28], the combination of a permutation-invariant base distribution with a permutation-equivariant diffeomorphism yields a permutation-invariant distribution q, as desired. More implementation details are provided in the SM.…”