Recent works have shown that generative models leave traces of their underlyinggenerative process on the generated samples, broadly referred to as fingerprints of agenerative model, and have studied their utility in detecting synthetic images fromreal ones. However, the extent to which these fingerprints can distinguish betweenvarious types of synthetic images and help identify the underlying generativeprocess remain under-explored. In particular, the very definition of a fingerprintremains unclear, to our knowledge. To that end, in this work, we formalize thedefinition of artifact and fingerprint in generative models, propose an algorithm forcomputing them in practice, and finally study how different design parameters affectthe model fingerprints and their attributability. We find that using our proposeddefinition can significantly improve the performance on the task of identifyingthe underlying generative process from samples (model attribution) compared toexisting methods. Additionally, we study the structure of the fingerprints andobserve that it is very predictive of the effect of different design choices on thegenerative process.