Variational Inference for SDEs Driven by Fractional Noise
Rembert Daems Manfred Opper Guillaume Crevecoeur Tolga Birdal
We present a novel variational framework for performing inference in (neural)
stochastic differential equations (SDEs) driven by Markov-approximate
fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling
real-world continuous-time dynamic systems with inherent noise and randomness.
Combining SDEs with the powerful inference capabilities of variational methods,
enables the learning of representative function distributions through
stochastic gradient descent. However, conventional SDEs typically assume the
underlying noise to follow a Brownian motion (BM), which hinders their ability
to capture long-term dependencies. In contrast, fractional Brownian motion
(fBM) extends BM to encompass non-Markovian dynamics, but existing methods for
inferring fBM parameters are either computationally demanding or statistically
inefficient. In this paper, building upon the Markov approximation of fBM, we
derive the evidence lower bound essential for efficient variational inference
of posterior path measures, drawing from the well-established field of
stochastic analysis. Additionally, we provide a closed-form expression to
determine optimal approximation coefficients. Furthermore, we propose the use
of neural networks to learn the drift, diffusion and control terms within our
variational posterior, leading to the variational training of neural-SDEs. In
this framework, we also optimize the Hurst index, governing the nature of our
fractional noise. Beyond validation on synthetic data, we contribute a novel
architecture for variational latent video prediction,-an approach that, to the
best of our knowledge, enables the first variational neural-SDE application to
video perception.