Latent diffusion models (LDMs) have emerged as a prominent framework for image generation, consisting of a diffusion model$\mathcal{M}$and a VAE decoder$\mathcal{D}$. High-quality image generation models are large and computationally intensive. As a result, image generation is typically performed on cloud servers, with the generated images then transmitted to edge devices.

Abstract We give a construction of the free dcpo-cone over any dcpo. There are two steps for getting this result. Firstly, we extend the notion of power domain to directed spaces which are equivalent to $T_0$ monotone-determined spaces introduced by Erné, and we construct the probabilistic powerspace of the monotone determined space, which is defined as a free monotone determined cone. Secondly, we take D-completion of the free monotone determined cone over the dcpo with its Scott topology. In addition, we show that generally the valuation power domain of any dcpo is not the free dcpo-cone.