latent-riemannian-world 0.1.0

Riemannian geometry and Bayesian inference for diffusion model latent spaces

latent-riemannian-world

Riemannian geometry + Bayesian inference + world models for diffusion model latent spaces.

Existing tools treat latent space as Euclidean. This library treats it as a Riemannian manifold — computing geodesics (true shortest paths on curved space), parallel transport, and curvature. Bayesian extensions add uncertainty quantification. World model extensions add temporal state transitions.

Why this library?

	latent-geometry	Diffusion-Pullback	lrw
Pullback metric	✓	✓	✓
Fisher-Rao metric	✗	✗	✓
Bayesian (uncertain) metric	✗	✗	✓
Geodesic solver	✗	✗	✓
Parallel transport	✗	✗	✓
SVGD / Riemannian SGLD	✗	✗	✓
World model / temporal	✗	✗	✓
Python 3.12+ / PyTorch 2.4+	✗	✗	✓
pip installable	✓	✗	✓

Installation

pip install latent-riemannian-world

Quick Start

import torch
from lrw.metric import PullbackMetric, BayesianMetric
from lrw.geodesic import GeodesicSolver
from lrw.transport import SchildsLadder
from lrw.bayes import SVGD, RiemannianSGLD
from lrw.world import LatentStateSpace, RiemannianRSSM

# Your diffusion model decoder
decoder = your_model.decode   # (B, D) -> (B, C, H, W)

# Pullback metric: G(z) = J(z)^T J(z)
metric = PullbackMetric(decoder=decoder)
z = torch.randn(4, 16)
G = metric.metric_tensor(z)   # (4, 16, 16) Riemannian metric matrices

# Geodesic interpolation between two latent points
solver = GeodesicSolver(metric=metric)
path = solver.interpolate(z[0:1], z[1:2], n_points=10)

# Parallel transport of a style vector along the geodesic
ladder = SchildsLadder(metric=metric)
v_transported = ladder.transport(z[0:1], z[1:2], style_vector)

# Bayesian metric with MC-Dropout ensemble
decoders = [decoder_with_dropout() for _ in range(8)]
bayes_metric = BayesianMetric(decoder_ensemble=decoders)
G_bayes = bayes_metric.metric_tensor(z)
G_var   = bayes_metric.metric_variance(z)

# World model: temporal transitions as geodesic flows
state_space = LatentStateSpace(metric=metric, dt=0.1)
z_next, v_next = state_space.step(z, velocity)
states, velocities = state_space.rollout(z, velocity, n_steps=20)

Module Structure

lrw/
├── metric/      PullbackMetric, FisherMetric, BayesianMetric
├── geodesic/    GeodesicSolver, slerp, slerp_path
├── transport/   SchildsLadder
├── bayes/       SVGD, RiemannianSGLD
├── world/       LatentStateSpace, RiemannianRSSM
└── utils/       sym_inv, sym_sqrt, riemannian_norm, batch_jacobian

References

• Shao et al. (2018) The Riemannian Geometry of Deep Generative Models. CVPR.
• Arvanitidis et al. (2018) Latent Space Oddity: on the Curvature of Deep Generative Models. ICLR.
• Park et al. (2023) Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry. NeurIPS.
• Liu et al. (2016) Stein Variational Gradient Descent. NeurIPS.
• Hafner et al. (2020) Dream to Control: Learning Behaviors by Latent Imagination. ICLR.

License

This project is licensed under the Business Source License 1.1 (BSL-1.1). Non-production use (research, personal projects, evaluation) is free. For commercial/production use, contact the author.

(c) 2025 lajjadred