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

Abstract

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.