Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds (2410.12779v4)

Abstract: Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold using geodesic-guided flows. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over the state-of-the-art methods in single-cell population-level trajectory inference.

• The paper presents a novel autoencoder that preserves manifold geometry via a distance matching loss.
• It introduces a warped Riemannian metric to enforce geodesic consistency, improving data generation and interpolation.
• Experimental results show a 30% improvement in trajectory inference for single-cell RNA sequencing datasets.

Geometry-Aware Generative Autoencoders for Manifold Learning and Generation

The paper introduces a novel framework known as Geometry-Aware Generative Autoencoder (GAGA), designed to address significant challenges inherent in high-dimensional data analysis. This framework combines manifold learning with generative modeling to facilitate the generation of data, interpolation along meaningful trajectories, and transportation across different populations, all while respecting the manifold structure of the data.

Contributions and Methodology

1. Autoencoder Architecture and Manifold Learning

The GAGA framework is built on the foundation of an autoencoder that respects the intrinsic geometry of high-dimensional data which often resides on low-dimensional manifolds. A primary contribution is the incorporation of a novel distance matching loss, facilitating the preservation of manifold distances in the latent space. The autoencoder leverages manifold learning techniques to embed data faithfully, ensuring that generated points adhere to the learned manifold's structure.

2. Warped Riemannian Metric

A key innovation in this work is the development of a warped Riemannian metric on the data space, which is crucial for geometry-aware data generation. This metric is learned by embedding both data points and negative samples, which are points off the manifold, to effectively characterize the geometry across the entire latent space. The warped metric imposes penalties for deviating from the manifold, thereby ensuring that geodesics remain within the data density.

3. Applications and Utility

GAGA demonstrates its capabilities in several complex data analysis tasks:

• Uniform Sampling: By utilizing a volume element derived from the warped metric, GAGA can generate points uniformly across the manifold. This capability is particularly useful for addressing data imbalance issues by evenly distributing generated points across sparsely-sampled areas.
• Interpolation and Geodesics: The autoencoder enables interpolation between two points on the manifold via geodesics, which is particularly useful for understanding transitions and progression in biological systems, such as cellular differentiation.
• Population Transport: The framework is also adapted for the dynamical optimal transport problem, allowing for effective transport of populations across different conditions. This is achieved by aligning starting and ending distributions and computing optimal, geodesic-guided flow paths.

Experimental Results and Comparisons

GAGA's performance was validated using both synthetic datasets, such as ellipsoids and tori, and real-world biological datasets, demonstrating substantial improvements over existing methods. Notably, GAGA achieved a 30% improvement in population trajectory inference for single-cell RNA sequencing data. The framework's ability to generate data that adheres closely to the manifold was shown to significantly mitigate issues with data sparsity and imbalance, as well as provide accurate geodesic interpolation.

Implications and Future Directions

The implications of this research are both practical and theoretical. Practically, the ability to generate data with a stronger adherence to underlying geometries holds promise for fields reliant on high-dimensional datasets, such as genomics and complex systems modeling. Theoretically, the framework advances the methodology of combining generative models with manifold learning, setting a precedent for future research that explores the intersection of geometry and data generation.

Looking ahead, the GAGA framework could inspire the development of more sophisticated methods that integrate geometric insights into deep learning architectures. Further research may explore extensions of the warped Riemannian metrics to other types of generative models, such as GANs and VAEs, potentially expanding the applicability of geometry-aware generative modeling in diverse scientific fields.

In summary, the development of GAGA represents a significant step forward in addressing the unique challenges posed by high-dimensional and manifold-structured data, offering a robust tool for both data generation and analysis.