When Do Local Score Models Extrapolate Across Size? A Diagnostic Theory and Benchmark

• URL: http://arxiv.org/abs/2606.09705v1
• Date: Mon, 08 Jun 2026 16:24:31 GMT
• Title: When Do Local Score Models Extrapolate Across Size? A Diagnostic Theory and Benchmark
• Abstract summary: We show that architectural locality alone does not guarantee stable size extrapolation.<n>We formalize this mechanism, proving a size-uniform comparison theorem for local marginals.<n> Empirically, we validate the interplay between spatial mixing, smoothed-score quasi-locality, and model receptive fields.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Scientific generative modeling often requires size transfer, where models trained on small systems are evaluated on larger ones. While translation-invariant architectures enable this evaluation, we show that architectural locality alone does not guarantee stable size extrapolation. Instead, stable extrapolation is governed by the quasi-locality of the Gaussian-smoothed score. Through Tweedie's formula, far-away perturbations can influence local score components via posterior covariance, meaning a local model succeeds only if its receptive field covers the smoothed score's response range. We formalize this mechanism, proving a size-uniform comparison theorem for local marginals under reverse diffusion. We also introduce Finite-Depth Local Flow (FDLF), a white-box diagnostic benchmark with exact scores, densities, and controllable response ranges. Empirically, we validate the interplay between spatial mixing, smoothed-score quasi-locality, and model receptive fields. Under spatial mixing, the smoothed score remains quasi-local relative to the receptive field, enabling stable extrapolation. Conversely, when spatial mixing weakens, the score's locality rapidly degrades, causing size transfer to fail.

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