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GMVAE (Poincaré)

📐 Gaussian Geometric

Hyperbolic Gaussian Mixture VAE

RNA

GMVAE in hyperbolic Poincaré space for hierarchical cell type relationships

Publications

Hyperbolic Deep Learning

Mathieu et al.2019
Complexity
★★★
complex
Interpretability
★★★
high
Architecture
Hyperbolic GMVAE
Latent Dim
10
Used in LAIOR Framework

Hyperbolic Hierarchical Clustering

Uses Poincaré ball model to represent hierarchical relationships between cell types naturally in hyperbolic geometry with full VAE reconstruction

Main Idea

Capture hierarchical cell type relationships using hyperbolic geometry that naturally represents tree structures

Key Components

Poincaré Encoder

Maps cells to Poincaré ball

Hyperbolic Distances

Measures similarity in hyperbolic space

Wrapped Normal Distribution

Gaussian analog in hyperbolic space for mixture

Geometric Decoder

Reconstructs expression from hyperbolic embeddings

Mathematical Formulation

z ∈ B^d (Poincaré ball); d_H(z_i,z_j) = acosh(1 + 2||z_i⊖z_j||²/((1-||z_i||²)(1-||z_j||²))); X̂ = Decoder(z)

Loss Functions

Hyperbolic ELBO
Reconstruction + KL in hyperbolic space

Data Flow

Expression → Poincaré Encoder → Hyperbolic Mixture → Geometric Decoder → Reconstructed Expression

Architecture Details

Architecture Type

GMVAE in Poincaré Ball (VAE Architecture)

Input/Output Types

single-cell → reconstruction

Key Layers

PoincareEncoderWrappedNormalGeometricDecoder

Frameworks

PyTorch

Tags

vaehyperbolichierarchicalgeometricgenerativerna