In forensic investigations of machine learning models, techniques that determine a model's data domain play an essential role, with prior work relying on large-scale corpora like ImageNet to approximate the target model's domain. Although such methods are effective in finding broad domains, they often struggle in identifying finer-grained classes within those domains. In this paper, we introduce an enhanced approach to determine not just the general data domain (e.g., human face) but also its specific attributes (e.g., wearing glasses). Our approach uses an image embedding model as the encoder and a generative model as the decoder. Beginning with a coarse-grained description, the decoder generates a set of images, which are then presented to the unknown target model. Successful classifications by the model guide the encoder to refine the description, which in turn, are used to produce a more specific set of images in the subsequent iteration. This iterative refinement narrows down the exact class of interest. A key strength of our approach lies in leveraging the expansive dataset, LAION-5B, on which the generative model Stable Diffusion is trained. This enlarges our search space beyond traditional corpora, such as ImageNet. Empirical results showcase our method's performance in identifying specific attributes of a model's input domain, paving the way for more detailed forensic analyses of deep learning models.