GMVAE (LearnablePGM)

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GMVAE with learnable curvature PGM for adaptive geometric structure

Publications

Geometric Manifold Learning with Adaptive Curvature for Single-Cell Analysis

GM-VAE Authors2024

Complexity

★★★

complex

Interpretability

★★☆

medium

Architecture

LearnablePGM-GMVAE

Latent Dim

Used in LAIOR Framework

Adaptive Curvature Geometric Clustering

LearnablePGM extends PGM by learning per-dimension curvature parameters (α, β², c), enabling adaptive geometric structure that adjusts to data characteristics

Main Idea

Learn both latent representations AND per-dimension geometric curvature, allowing the manifold to adaptively adjust its structure from data

Key Components

ExpEncoderLayer

Maps features to half-plane via exponential map from Poincaré disk; encodes learnable [α, log(β²), log(c)]

Learnable Curvature Parameters

[α, log(β²), log(c)] per latent dimension: shape parameter α, variance log(β²), curvature magnitude log(c)

Adaptive Manifold

Geometric structure adapts per latent dimension via learned c parameter

LogDecoderLayer

Maps latent from half-plane to Poincaré disk via logarithmic map; reconstructs expression

Mathematical Formulation

θ_i = [α_i, β_i², c_i] learned per dimension i; manifold curvature = -c_i; z ~ PGM(θ); X̂ = Decoder(z)

Loss Functions

LearnablePGM-ELBO

E_q[log p(x|z)] - KL(q(z|α,β²,c) || p(z|α_0,β_0²,c_0))

Data Flow

Expression → ExpEncoder [α,β²,c] → Adaptive Manifold Space → LogDecoder → Reconstructed Expression

Architecture Details

Architecture Type

GMVAE with Learnable Per-Dimension Curvature (VAE Architecture)

Input/Output Types

single-cell → reconstruction

Key Layers

ExpEncoderLayerLearnableManifoldLogDecoderLayerGeooptPoincareBall

Frameworks

PyTorchgeoopt

Learnable Pseudo-Gaussian Manifold VAE