Lorentz-regularized interpretable VAE for multi-scale single-cell transcriptomic and epigenomic embeddings.

Background: Single-cell multi-omics technologies capture cellular heterogeneity at unprecedented resolution, yet dimensionality reduction methods face a fundamental local-global trade-off: approaches optimized for local neighborhood preservation distort global topology, while those emphasizing global coherence obscure fine-grained cell states. Results: We introduce the Lorentz-regularized variational autoencoder (LiVAE), a dual-pathway architecture that applies hyperbolic geometry as soft regularization over standard Euclidean latent spaces. A primary encoding pathway preserves local transcriptional details for high-fidelity reconstruction, while an information bottleneck (BN) pathway extracts global hierarchical structure by filtering technical noise. Lorentzian distance constraints enforce geometric consistency between pathways in hyperbolic space, enabling LiVAE to balance local fidelity with global coherence without requiring specialized batch-correction procedures. Systematic benchmarking across 135 datasets against 21 baseline methods demonstrated that LiVAE achieves superior global topology preservation (distance correlation gains: 0.209-0.436), richer latent geometry (manifold dimensionality: 0.123-0.467; participation ratio: 0.149-0.761), and enhanced robustness (noise resilience: 0.184-0.712) while maintaining competitive local fidelity. The overall embedding quality improved by 0.051-0.284 across uniform manifold approximation and projection (UMAP) and t-distributed stochastic neighbor embedding (t-SNE) visualizations. Component-wise interpretability analysis on a Conclusion: LiVAE provides a robust, general-purpose framework for single-cell representation learning that resolves the local-global trade-off through geometric regularization. By maintaining Euclidean latent spaces while leveraging hyperbolic priors, LiVAE enables improved developmental trajectory inference and mechanistic biological discovery without sacrificing compatibility with existing computational ecosystems.