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Stable Diffusion

Stability AI

Structures found in this family (6)

Score-Jacobian pullback Riemannian metric, Linear Direction, Linear Subspace, Intrinsic-dimension profile across depth, Hyperbolic Manifold, Curvature profile of the representation manifold

By model (10)

Stable Diffusion XL
Linear Direction, Intrinsic-dimension profile across depth, Hyperbolic Manifold
Stable Diffusion v1.4
Linear Subspace, Linear Direction, Intrinsic-dimension profile across depth
Stable Diffusion v1.5
Score-Jacobian pullback Riemannian metric, Linear Direction, Linear Subspace
Stable Diffusion 3.5
Hyperbolic Manifold, Linear Direction
SDXL-DMD (4-step distilled)
Intrinsic-dimension profile across depth, Linear Direction
Stable Diffusion v2.1
Linear Direction, Linear Subspace
Stable Diffusion 1.5 (Dreamshaper8 checkpoint)
Curvature profile of the representation manifold
Stable Diffusion 3
Hyperbolic Manifold
SDXL Turbo
Linear Direction
Stable Diffusion 3.5 Medium
Curvature profile of the representation manifold

Observations (21)

A Riemannian metric derived directly from Stable Diffusion's own score-function Jacobian -- not pulled back from any assumed linear latent -- and geodesic-guided interpolation outperforms Lerp/Slerp baselines
A supervised SAE gives one latent per concept for erasure
Diffusion concepts occupy provably-separate, editable score subspaces
Concept Sliders are LoRA weight-space directions for diffusion control
A pullback Riemannian metric yields diffusion tangent-space directions
DINOv2 and Stable Diffusion encode depth/normals via nonlinear dense probes
Concept-direction local ID tracks generality and restores distilled-diffusion diversity
Hyperbolic parallel-transport steering outperforms Euclidean concept control
A real Stable Diffusion 1.5 latent space shows a fractal structure of Fisher-metric phase transitions, with diverging Lipschitz constants at phase boundaries
Activation-derived forget/retain subspaces edited into diffusion weights
SAE features from SDXL-Turbo's one-step U-Net transfer zero-shot and edit images
SD concept directions show composition, style, and texture set at successive denoising steps
A real DDPM's Critical Boundary Detector correlates with LPIPS and localizes classifier-guidance-worthy timesteps
A diffusion score-Jacobian splits into tangent and normal subspaces
Self-supervised h-space directions enable responsible diffusion generation
SliderSpace realizes CLIP-PCA directions as causally independent LoRA sliders
Stable Diffusion's intrinsic dimension tracks prompt perplexity in bottleneck layers
OCE erases diffusion concepts via a shared orthogonal rotation preserving norms and angles
A single RFM/AGOP direction per concept steers diffusion at a fraction of guidance cost
Position-aware SAE safety steering generalizes across domains within a diffusion model
Residualizing adjacent denoising timesteps yields directionally-stable SD feature trajectories

Papers

Image Interpolation with Score-based Riemannian Metrics of Diffusion Models (2025), SAEmnesia: Erasing Concepts in Diffusion Models with Supervised Sparse Autoencoders (2025), Concept Algebra for (Score-Based) Text-Controlled Generative Models (2023), Concept Sliders: LoRA Adaptors for Precise Control in Diffusion Models (2023), Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry (2023), Probing the 3D Awareness of Visual Foundation Models (2024), ELROND: Exploring and Decomposing Intrinsic Capabilities of Diffusion Models (2026), Not All Latent Spaces Are Flat: Hyperbolic Concept Control (2026), Hessian Geometry of Latent Space in Generative Models (2026), Concept Unlearning via Cross-Attention Activation Projection for Diffusion Models (2026), One-Step Is Enough: Sparse Autoencoders for Text-to-Image Diffusion Models (2024), Emergence and Evolution of Interpretable Concepts in Diffusion Models (2025), The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics (2026), Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models (2025), Self-Discovering Interpretable Diffusion Latent Directions for Responsible Text-to-Image Generation (2024), SliderSpace: Decomposing the Visual Capabilities of Diffusion Models (2025), Exploring the Representation Manifolds of Stable Diffusion Through the Lens of Intrinsic Dimension (2023), Orthogonal Concept Erasure for Diffusion Models (2026), General and Efficient Steering of Diffusion Models (2026), Robust and Generalizable Safety Steering for Text-to-Image Diffusion Transformers (2026), Residualized Temporal Sparse Autoencoders for Interpreting Diffusion Models (2026)