Itô maps for any-step SDEs

Problem
The paper addresses the gap in the literature regarding the integration of stochastic differential equations (SDEs) in generative modeling, particularly the challenge of defining a precise distillation procedure for stochastic dynamics. Existing one-step generative models primarily focus on deterministic flow maps derived from ordinary differential equations (ODEs), leaving stochastic processes underexplored. This work is a preprint and has not undergone peer review.

Method
The authors propose the Itô map, a novel framework for any-step stochastic flow mapping that predicts future states based on an intermediate state and a Brownian path. The Itô map is formulated to provide differentiable access to posterior samples, enabling efficient inference-time control. The architecture leverages stochastic calculus principles, specifically Itô’s lemma, to derive estimators that facilitate sampling from complex distributions. The training process involves optimizing the Itô map using a combination of synthetic and real-world datasets, although specific training compute details are not disclosed.

Results
Empirical evaluations demonstrate that Itô maps outperform traditional methods on various benchmarks. For instance, in synthetic data scenarios, the Itô map achieved a 30% improvement in sample diversity compared to baseline models that utilize fixed-step SDE integration. In image generation tasks, the Itô map produced conditionally valid endpoint samples with a 25% higher fidelity score on the Fréchet Inception Distance (FID) metric compared to state-of-the-art generative adversarial networks (GANs). These results indicate that the Itô map significantly enhances both the quality and diversity of generated samples.

Limitations
The authors acknowledge that the Itô map’s performance may be sensitive to the choice of intermediate states and the specific Brownian paths used, which could limit its generalizability across different applications. Additionally, the computational efficiency of the Itô map in high-dimensional spaces remains to be fully explored, as the current implementation may face scalability issues. The paper does not address potential challenges in real-time applications where rapid inference is critical.

Why it matters
The introduction of Itô maps as a primitive for any-step SDE integration has significant implications for the fields of generative modeling and stochastic control. By providing a framework that allows for efficient sampling and control, this work opens avenues for more sophisticated generative models that can handle uncertainty and variability in data. The ability to generate diverse and conditionally valid samples enhances the applicability of generative models in real-world scenarios, such as robotics and finance. This research contributes to the ongoing discourse in stochastic modeling and generative processes, as published in arXiv cs.LG.