Under Ahlfors regularity of exponent β, the minimal energy distance between a measure and its N-point empirical version decays exactly as N to the power -½(1 + q/β) for power kernels with exponent q in (0,2).
Borgwardt, Malte J
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Causality-encoded diffusion models use a known DAG to train graph-consistent conditional diffusions for observational recovery, interventional sampling via fixed-variable propagation, and a resampling-based directed edge test with convergence rates depending on local dimension.
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Sharp Rates of MMD Empirical Estimation with Power Kernels
Under Ahlfors regularity of exponent β, the minimal energy distance between a measure and its N-point empirical version decays exactly as N to the power -½(1 + q/β) for power kernels with exponent q in (0,2).
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Causality-Encoded Diffusion Models for Interventional Sampling and Edge Inference
Causality-encoded diffusion models use a known DAG to train graph-consistent conditional diffusions for observational recovery, interventional sampling via fixed-variable propagation, and a resampling-based directed edge test with convergence rates depending on local dimension.