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On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows
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We present a novel theoretical framework for understanding the expressive power of normalizing flows. Despite their prevalence in scientific applications, a comprehensive understanding of flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. We propose a distributional universality theorem for well-conditioned coupling-based normalizing flows such as RealNVP. In addition, we show that volume-preserving normalizing flows are not universal, what distribution they learn instead, and how to fix their expressivity. Our results support the general wisdom that affine and related couplings are expressive and in general outperform volume-preserving flows, bridging a gap between empirical results and theoretical understanding.
Forward citations
Cited by 7 Pith papers
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MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation
MIMFlow uses a VAE on masked images to feed semantic latents to a normalizing flow while a decoder handles high-frequency details, reporting FID 2.50 and 71.3% linear probing on ImageNet 256x256 with 128 tokens.
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Characterizing Neural Manifolds' Properties and Curvatures using Normalizing Flows
A normalizing flow with a mixture-of-Gaussians latent space and a quadratic post-hoc approximation yields higher-order correlations and curvature estimates for neural manifolds in macaque visual cortex.
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Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows
A conditional normalizing flow method that solves the primal optimal transport problem and computes Wasserstein-2 barycenters as weighted averages of maps from a shared latent distribution.
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Expert-elicitation method for non-parametric joint priors using normalizing flows
A normalizing-flow-based method learns non-parametric joint prior distributions for Bayesian models by matching simulated predictions to expert-elicited statistics.
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Neural Conjugate Flows: Physics-informed architectures with flow structure
Neural Conjugate Flows represent ODE solutions as a learned coordinate change followed by an affine flow, giving exact group structure and a universal approximation claim.
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Normalizing Flows are Capable Models for Continuous Control
A simple normalizing flow policy matches or outperforms diffusion and autoregressive baselines across imitation learning, offline RL, goal-conditioned RL, and unsupervised RL on 82 tasks.
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Causally Consistent Normalizing Flow
A normalizing flow that preserves causal structure by transforming variables in topological batches, enabling deep causally consistent generative models.
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