Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime

What the math gives to ML

The paper develops finite-size fluctuation control for mean-field particle systems by centering a pair interaction against a background law and expressing the residual quadratic fluctuations through a Carleman–Fredholm determinant. The transferable asset is the Hilbert–Schmidt spectral structure: square-summable eigenvalues permit a stable, trace-corrected normalization even when the raw interaction is not trace class. This can be adapted to neural representation losses that use pairwise repulsion, diversity, or similarity kernels, reducing sensitivity to batch size and suppressing unstable spectral directions. A practical implementation uses a centered kernel matrix, a truncated eigendecomposition, and a determinant correction computed from a small landmark buffer.