Metric Flow Matching for Smooth Interpolations on the Data Manifold (2405.14780v2)

Abstract: Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of Euclidean geometry, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.

• The paper introduces Metric Flow Matching to overcome Euclidean limitations by using a learned Riemannian metric for natural trajectory interpolations.
• It employs a geometric loss function that minimizes geodesic deviation, ensuring results remain on the intrinsic data manifold.
• Empirical evaluations show enhanced performance in trajectory inference, LiDAR navigation, and image translation compared to traditional methods.

Metric Flow Matching for Smooth Interpolations on the Data Manifold

Overview

The paper introduced by Kapusniak et al. discusses a novel approach titled Metric Flow Matching (MFM) for addressing specific limitations in generative models, particularly within the framework of Conditional Flow Matching (CFM). Generative models traditionally rely on interpolations between source and target distributions under Euclidean geometry assumptions. However, straight-line interpolations in Euclidean space can deviate from the underlying data manifold, potentially leading to inaccurate or unnatural matches. MFM addresses this by leveraging a data-dependent Riemannian metric to induce interpolations that conform more closely to the data manifold.

Problem Context and Motivation

For various scientific and natural domains such as single-cell RNA sequencing or image translation, one often needs to infer system dynamics from sparse, static measurements. The challenge lies in constructing conditional paths that effectively transform a source distribution (e.g., a protein expression profile in cells) into a target distribution. The conventional approach in generative models employs Euclidean geometry, resulting in straight-line interpolations which may not capture the underlying data manifold's nonlinear dynamics. This discrepancy poses significant issues, particularly in trajectory inference tasks where the inferred paths should ideally remain on the data manifold to ensure meaningful reconstructions.

Contributions and Approach

The authors propose Metric Flow Matching (MFM) as a simulation-free framework that adapts Conditional Flow Matching to the Riemannian manifold defined by a data-dependent metric. Here are the primary contributions of the paper:

1. Data-Dependent Metric Design: MFM tackles the geometric discrepancy by utilizing a learned Riemannian metric. This metric assigns lower costs to regions with high data concentration, guiding the interpolants to stay on the data manifold.
2. Geometric Loss Function: The interpolants are learned by minimizing a geodesic loss, which penalizes the interpolant's velocity based on the metric. This approach ensures that the generated paths are approximate geodesics of the data-induced metric.
3. General Framework: The authors prescribe a family of metrics independent of specific tasks, including Radial Basis Function (RBF) networks, which are adaptable to different datasets and applications.
4. Empirical Validation: The paper validates MFM on multiple tasks, showing significant improvements in trajectory inference for single-cell dynamics, navigation within LiDAR point clouds, and unpaired image translations.

Numerical Results and Experimental Validation

Kapusniak et al. demonstrate the efficacy of MFM through a series of empirical evaluations. Key results include:

1. Trajectory Inference: On single-cell RNA sequencing data, MFM significantly outperformed Euclidean CFM and various baselines such as Schrödinger Bridge models and optimal transport methods. Specifically, MFM achieved superior results on metrics like the Wasserstein distance across multiple datasets.
2. LiDAR Navigation: The interpolants learned via MFM resulted in more natural and meaningful trajectories on complex surfaces scanned by LiDAR, compared to straight-line interpolants derived from Euclidean CFM.
3. Unpaired Image Translation: In translations within the AFHQ dataset, MFM not only improved image quality, as demonstrated by better FID and LPIPS scores, but also preserved input features more effectively by ensuring smoother interpolations on the underlying data manifold.

Implications and Future Directions

The introduction of MFM opens several avenues for theoretical and practical advancement:

• Richer Interpolations: By incorporating data-dependent metrics, MFM enables richer and more accurate interpolations, crucial for applications needing high fidelity in trajectory predictions.
• Simulation-Free Generalization: MFM's formulation avoids the need for computationally expensive simulations, making it scalable to high-dimensional data and tasks.
• Applicability to Different Domains: The flexibility in metric design allows MFM to be adapted across various data types and tasks, from biomedical applications to image processing.

Future developments might explore:

• Task-Specific Metric Design: While the current metrics are task-agnostic, designing metrics that incorporate domain-specific biases could enhance the performance further.
• Extension to Other Generative Models: The principles underlying MFM could extend beyond CFM, potentially benefiting score-based generative models like diffusion models.
• Non-Euclidean Ambient Spaces: Exploring methods to learn interpolants in non-Euclidean ambient spaces might yield even more general and robust models.

Conclusion

Metric Flow Matching represents a substantial advancement in generative modeling by addressing the core issue of interpolations straying off the data manifold. The paper's contributions highlight the importance of geometry-aware pathways in generative models, enabling them to generate more accurate, reliable, and meaningful outputs across various applications.