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Variational Inference for SDEs Driven by Fractional Noise

ICLR 2024 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Annealing and divergence-based variational objectives · 2 citations (OpenAlex)

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Proposes variational inference for neural SDEs driven by Markov-approximate fractional Brownian motion, with closed-form optimal OU combination weights, and applies it to latent video prediction on Stochastic Moving MNIST.

Problem

Inference in SDEs driven by fractional Brownian motion (non-Markovian noise with Hurst index) is computationally hard, and existing methods are costly or statistically inefficient; standard SDE variational tools assume Brownian noise.

Approach

Approximate fBM as a finite linear combination of Markov Ornstein-Uhlenbeck processes driven by a common Wiener process, augment the state so the system is an ordinary SDE, derive an ELBO via Girsanov's theorem with a neural control term, give closed-form optimal combination weights minimizing L2 error, and learn the Hurst index by maximizing the ELBO.

Key results

• On Stochastic Moving MNIST, the MA-fBM latent SDE reaches PSNR 15.30 and ELBO -608.00, versus 14.90 and -913.60 for the BM-driven baseline, and is comparable to SVG (14.50 PSNR) though below SLRVP (16.93 PSNR).
• The learned Hurst index converged to H=0.90 (long-term memory) for the video model.
• The closed-form optimal omega_k gives considerably lower approximation error than a piecewise-Laplace baseline in the MSE-vs-K study, with diminishing returns as K grows.

Measured effects reported in the paper

Surprising findings

• The Markov approximation error saturates after a modest number of OU processes (K), with diminishing returns especially for low H.
• The authors note that Euler integration quickly deviates from analytical results for state-dependent diffusions, so Stratonovich-Milstein solvers are needed.

Limitations

• Increased computational overhead for long time horizons due to SDE integration.
• The Markov approximation lacks a tight convergence bound; only empirical super-polynomial convergence is observed.
• Video model is evaluated only on Stochastic Moving MNIST and only compared to a BM baseline ablation.

Experimental setup

Datasets: Stochastic Moving MNIST synthetic fractional OU bridge data

Benchmarks: Stochastic Moving MNIST fractional OU bridge posterior variance Hurst index estimation toy problem

Metrics: PSNR ELBO MSE posterior variance

Baselines: SVG SLRVP BM-driven latent SDE fractional neural SDE (Tong et al.)

Builds on: Girsanov theorem Markov representation of fBM latent SDE Stochastic Video Generation (SVG)

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