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Conditioning non-linear and infinite-dimensional diffusion processes

NeurIPS 2024 · Image generation › Fast diffusion sampling · approach: Diffusion theory and learned samplers · 3 citations (OpenAlex)

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Derives an infinite-dimensional Doob h-transform via Girsanov/Itô to condition non-linear Hilbert-space SDEs on exact or noisy endpoint observations, then discretises with a Fourier basis and learns the score by score matching for butterfly shape time series.

Problem

Conditioning non-linear, function-valued (infinite-dimensional) stochastic processes on observations at a terminal time, which has not been treated before; existing methods discretise first and then condition.

Approach

Extends Doob's h-transform to infinite-dimensional strong-solution SDEs using infinite-dimensional Itô and Girsanov theorems, so conditioning is done before discretisation; the resulting score term is discretised via a truncated Fourier basis and learned with score matching.

Key results

• On Brownian motion conditioning, Fourier-basis score learning with 8 bases reached RMSE 5.09 (100 epochs), versus 7.95 for landmarks with 8 points.
• Error grows with dimension: Fourier RMSE rose to 10.54 at 32 bases (300 epochs), and landmark RMSE was 10.79 at 32 points.
• Demonstrated conditioned butterfly shape trajectories between two Papilio species using a stochastic LDDMM-style SDE discretised via Fourier basis.

Measured effects reported in the paper

Surprising findings

• Learning score error does not decrease monotonically with dimension; higher-dimensional discretisations need more training epochs to keep errors low.

Limitations

• Theory relies on strong solutions and Itô's formula, so it does not directly extend to non-strong solutions.
• Only the time reversal is learned; errors compound when learning the forward bridge.
• Neural architecture did not scale well beyond moderate dimensions; error increases with number of bases.

Experimental setup

Datasets: Papilio butterfly outline dataset (segmented from gbif.org images)

Benchmarks: closed-form Brownian motion score

Metrics: RMSE training time

Baselines: landmark discretisation

Builds on: Doob's h-transform Girsanov's theorem score matching for diffusion bridges LDDMM

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