MMD

Under weight-free gradient flow, any stationary point whose MMD value is far from zero is unstable, and gradient flow will not stably remain at these bad stationary points.

MMD (maximum mean discrepancy) is a classic measure in machine learning for measuring the difference between two distributions, widely used in generative models, domain adaptation, and kernel two-sample tests.

If and only if the kernel is a characteristic kernel (such as a Gaussian kernel), MMD(P,Q)=0 is equivalent to P=Q.

MMD is widely applied in tasks such as the training objective of Generative Adversarial Networks (GANs), two-sample hypothesis testing, and domain adaptation.

When the kernel is a characteristic kernel (e.g., a Gaussian kernel), MMD equals zero if and only if the two distributions are identical.

The contractivity under kernel-induced geometry enters the convergence bound only through a linear correction term, rather than the dominant term.

The paper's experiments cover smooth soft-particle energy, maximum mean discrepancy (MMD) minimization, component ablation experiments, and scaling studies in both the number of particles and dimension directions.

Common UDA technical approaches include adversarial training (e.g., the DANN method), Maximum Mean Discrepancy (MMD), and self-training/pseudo-labeling

Domain adaptation assumes access to unlabeled data from the target domain and aligns the feature distributions of the source and target domains through adversarial training (such as the DANN gradient reversal layer), MMD minimization, or optimal transport methods.