Define latent spaces by example: optimisation over the outputs of generative models

Modern generative AI models such as diffusion and flow matching can sample from rich data distributions, but many downstream tasks -- such as experimental design or creative content generation -- require a higher level of control than unconstrained sampling. The challenge is to efficiently identify outputs that are both probable under the model and satisfy task-specific constraints. We address this by introducing surrogate latent spaces: non-parametric, low-dimensional Euclidean embeddings that can be extracted from any generative model without additional training. The axes in the Euclidean space can be defined via examples, providing a simple and interpretable approach to define custom latent spaces that both express intended features and are convenient to use in downstream tasks. The representation is Euclidean and has controllable dimensionality, permitting direct application of standard optimisation algorithms to traverse the outputs of generative models. Our approach is architecture-agnostic, incurs almost no additional computational cost, and generalises across modalities, including images, audio, videos, and structured objects like proteins.