Identifiability of interaction kernels in mean-field equations of interacting particles

We provide a complete characterization of the identifiability of interaction kernels in mean-field equations for interacting particle systems. The key is to identify function spaces in which a probabilistic quadratic loss functional has a unique minimizer. We consider two data adaptive $L^2$ spaces, one with the Lebesgue measure and the other with an exploration measure intrinsic to the mean-field equation. For each $L^2$ space, the Fr\'echet derivative of the loss functional leads to a semi-positive integral operator, thus, the identifiability holds on the eigen-spaces with nonzero eigenvalues of the integral operator, and the function space of identifiably is the $L^2$-closure of the RKHS related to the integral operator. Furthermore, the identifiability holds on the $L^2$ spaces if and only if the integral operators are strictly positive. Thus, the inverse problem is ill-posed and regularization is necessary. In the context of truncated SVD regularization, we demonstrate numerically that the weighted $L^2$ space is preferable over the unweighted $L^2$ space because it leads to more accurate regularized estimators.