Score matching for bridges without time-reversals (2407.15455v2)

Abstract: We propose a new algorithm for learning a bridged diffusion process using score-matching methods. Our method relies on reversing the dynamics of the forward process and using this to learn a score function, which, via Doob's $h$-transform, gives us a bridged diffusion process; that is, a process conditioned on an endpoint. In contrast to prior methods, ours learns the score term $\nabla_x \log p(t, x; T, y)$, for given $t, Y$ directly, completely avoiding the need for first learning a time reversal. We compare the performance of our algorithm with existing methods and see that it outperforms using the (learned) time-reversals to learn the score term. The code can be found at https://github.com/libbylbaker/forward_bridge.

• The paper presents a novel direct score matching method to learn the score function for conditioned diffusion processes without requiring time-reversal.
• It leverages adjoint process simulations and the Euler-Maruyama method to reduce approximation errors and computational overhead.
• Experiments on models like Ornstein-Uhlenbeck, Brownian motion, and cell differentiation demonstrate improved efficiency and accuracy over previous methods.

Score Matching for Bridges Without Time-Reversals

The paper "Score Matching for Bridges Without Time-Reversals" by Elizabeth L. Baker, Moritz Schauer, and Stefan Sommer introduces a novel algorithm for learning bridged diffusion processes. This method leverages score matching techniques to directly learn the score term $\nabla_x \log p(t, x; T, y)$, bypassing the need for time-reversal learning, which is traditionally used in such processes.

Methodology

The core contribution of the paper consists of a new approach for the construction of conditioned (or bridged) stochastic differential equations (SDEs). Traditional methods hinge on Doob's $h$-transform to condition diffusion processes on endpoint values. However, computing the required transition density term, $\nabla_x \log p(t, x; T, y)$, is generally intractable.

The authors propose using adjoint processes to directly learn this score term. Instead of learning a time-reversed process and then re-learning in the forward direction as in the method by \citet{heng_simulating_2022}, the new method learns the score term through simulated trajectories of adjoint processes, which represent the reversed dynamics but not the full time-reversal. This approach reduces approximation errors and computational overhead, as it avoids the double training step required in prior methods.

The key steps are:

1. Define the Adjoint Process: The adjoint process $\{Y(t), \mathcal{Y}(t)\}$ captures the reversed dynamics of the original process.
2. Score Matching Loss Function: The paper introduces a loss function that integrates over the adjoint process's trajectories to match the learned score function $s_\theta(t, x)$ with the true score.
3. Training the Neural Network: By simulating trajectories of the adjoint process using the Euler-Maruyama method and optimizing the loss function, the neural network learns the score $\nabla \log p(t, x; T, y)$.

Comparative Analysis

The proposed method shows significant improvements over existing techniques, particularly in scenarios where time-reversals are complex or non-linear. The authors compare their results with those obtained using methods by \citet{heng_simulating_2022}, demonstrating lower error rates and more efficient training.

Experiments include:

• Ornstein-Uhlenbeck Process: The new method's error in learning the score function is substantially lower compared to the time-reversal-based methods.
• Brownian Motion with Distributed Endpoints: Conditioning on endpoints distributed on a circle showcases the flexibility of the proposed approach.
• Cell Differentiation Model: Simulating critical biological processes, the method effectively learns the score, producing plausible and accurate conditioned trajectories.

Practical and Theoretical Implications

This research has several practical implications. For fields that rely on conditioned diffusion processes, such as physics, finance, and biology, the proposed method offers a more efficient and accurate way to construct these processes. The direct learning of the score function significantly reduces computational complexity, making the approach more scalable for high-dimensional systems.

Theoretically, this method opens new avenues in the understanding and application of diffusion processes, particularly in situations where the dynamics are complicated, and transition densities are not explicitly known. By sidestepping the need for time-reversals, it brings robustness and simplicity to the modeling and simulation of conditioned SDEs.

Future Developments

The paper suggests potential future developments, including extending the framework to handle more complex types of endpoint distributions and further optimizing the computational methods involved in simulating the adjoint processes. Advanced neural network architectures and learning algorithms could also be explored to enhance the efficiency and applicability of the method.

In summary, the paper "Score Matching for Bridges Without Time-Reversals" provides a significant contribution to the field of score matching for diffusion processes. By eliminating the necessity of learning time-reversed processes, it introduces a more streamlined and efficient approach to conditioned SDEs, with broad implications for both practical applications and theoretical advancements in stochastic modeling.