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Inflationary Flows: Calibrated Bayesian Inference with Diffusion-Based Models

NeurIPS 2024 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Simulation-based inference · 1 citations (OpenAlex)

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Repurposes diffusion-based models' probability flow ODE with novel inflationary noise and rescaling schedules to build a deterministic, invertible, dimension-preserving or dimension-reducing map from data to a Gaussian latent space, enabling calibrated posterior inference.

Problem

Variational and flow-based latent models yield overconfident or non-identifiable posteriors, and diffusion models do not compress data or preserve local structure, so calibrated Bayesian inference in a low-dimensional latent space is hard.

Approach

Define inflationary flows as pfODEs derived from Gaussian-convolution marginals with covariance schedule C(t) (GG^T = C-dot) plus a time-dependent coordinate rescaling A(t); choose PR-preserving or PR-reducing schedules so the ODE maps data deterministically and invertibly to a stationary Gaussian, with the score learned by standard DBM training.

Key results

• On AFHQv2 at 1.02 inflation gap, PR-reducing flow to 2 dimensions (~0.07% of nominal dimension) achieves FID 11.95 with round-trip MSE 1.55, while the 3072-dimensional case reaches FID 11.90.
• On CIFAR-10 at 30, 40, and 62 preserved dimensions, inflationary flows (IG=1.02) achieve FID 23.3, 24.3, and 23.2, versus 280.9 to 544.0 for M-Flow, Rectangular Flows and Canonical Manifold Flows.
• HMC posterior over GMM mixture weights in a 2D circles toy example covers ground-truth weights [0.5, 0.25, 0.25].

Measured effects reported in the paper

Surprising findings

• FID degrades sharply as the inflation gap grows (from 11.95 at IG=1.02 to 107.64 at IG=1.50 for 2 dimensions), showing a tradeoff between compression and generation quality.
• Compressed flows reach near-full-dimensional FID, e.g. 2 preserved dimensions on AFHQv2 match the 3072-dimensional model's FID.

Limitations

• Relies on the participation ratio, a second-order dimensionality measure, which favors top principal components when reducing dimension.
• Requires training diffusion models over much larger ranges of noise than typical, causing compression tradeoffs as inflation gap and preserved dimensions vary.

Experimental setup

Datasets: CIFAR-10 AFHQv2

Benchmarks: CIFAR-10 AFHQv2

Metrics: FID round-trip MSE

Baselines: M-Flows Rectangular Flows Canonical Manifold Flows

Builds on: probability flow ODE (Song et al.) EDM preconditioning (Karras et al.) flow matching Gaussian paths injective flows

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