  The last decade has witnessed the success of deep learning and the surge of
publicly released trained models, which necessitates the quantification of the
model functional distance for various purposes. However, quantifying the model
functional distance is always challenging due to the opacity in inner workings
and the heterogeneity in architectures or tasks. Inspired by the concept of
"field" in physics, in this work we introduce Model Gradient Field (abbr.
ModelGiF) to extract homogeneous representations from the heterogeneous
pre-trained models. Our main assumption underlying ModelGiF is that each
pre-trained deep model uniquely determines a ModelGiF over the input space. The
distance between models can thus be measured by the similarity between their
ModelGiFs. We validate the effectiveness of the proposed ModelGiF with a suite
of testbeds, including task relatedness estimation, intellectual property
protection, and model unlearning verification. Experimental results demonstrate
the versatility of the proposed ModelGiF on these tasks, with significantly
superiority performance to state-of-the-art competitors. Codes are available at
https://github.com/zju-vipa/modelgif.
