# A Geometric Unification of Generative AI with Manifold-Probabilistic Projection Models

arXiv: https://arxiv.org/abs/2510.00666 · HTML: https://mlsanity.com/paper/2510.00666/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 20 of its 27 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 20 of 20)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DiffBIR: Towards Blind Image Restoration with Generative Diffusion Prior](https://mlsanity.com/paper/2308.15070/) (2023)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [On the Trajectory Regularity of ODE-based Diffusion Sampling](https://mlsanity.com/paper/2405.11326/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Is Noise Conditioning Necessary for Denoising Generative Models?](https://mlsanity.com/paper/2502.13129/) (2025)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [SCUT-FBP5500: A Diverse Benchmark Dataset for Multi-Paradigm Facial Beauty Prediction](https://mlsanity.com/paper/1801.06345/) (2018)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)

## Abstract (arXiv)

Most models of generative AI for images assume that images are inherently low-dimensional objects embedded within a high-dimensional space. Additionally, it is often implicitly assumed that thematic image datasets form smooth or piecewise smooth manifolds. Common approaches overlook the geometric structure and focus solely on probabilistic methods, approximating the probability distribution through universal approximation techniques such as the kernel method. In some generative models the low dimensional nature of the data manifest itself by the introduction of a lower dimensional latent space. Yet, the probability distribution in the latent or the manifold's coordinate space is considered uninteresting and is predefined or considered uniform. In this study, we address the problem of Blind Image Denoising (BID), and to some extent, the problem of generating images from noise by unifying geometric and probabilistic perspectives. We introduce a novel framework that improves upon existing probabilistic approaches by incorporating geometric assumptions that enable the effective use of kernel-based probabilistic methods. Furthermore, the proposed framework extends prior geometric approaches by combining explicit and implicit manifold descriptions through the introduction of a distance function. The resulting framework demystifies diffusion models by interpreting them as a projection mechanism onto the manifold of ``good images''. This interpretation leads to the construction of a new deterministic model, the Manifold-Probabilistic Projection Model (MPPM), which operates in both the representation (pixel) space and the latent space. We demonstrate that the Latent MPPM (LMPPM) outperforms the Latent Diffusion Model (LDM) across various datasets, achieving superior results in terms of image restoration and generation.