Dimension-Free Scaling Laws for Invariant Score Matching

Modern machine learning models increasingly operate on variable-size data, such as sets and graphs, where the input dimension may differ between training and testing. Classical estimation theory is largely tied to fixed domains, with guarantees that often deteriorate as dimension grows. We ask whether invariant score matching obeys dimension-free scaling laws: can a score estimator learned on one domain transfer reliably to another, possibly much larger, domain? We answer this question by developing a general theory of invariant $M$-estimation across dimensions, with score matching as a central instance. Our analysis introduces a new spectral complexity measure that governs the scaling law of the estimator by quantifying how efficiently the target score is captured by low-complexity invariant components. For score matching on sets and graphs, we prove dimension-uniform convergence guarantees and show that low-complexity invariant scores can be estimated at fast, dimension-free rates, nearly matching the parametric rate. Our theory identifies a sharp scaling transition: below an explicit spectral threshold, invariant score matching transfers across dimensions with fast rates independent of ambient dimension; above this threshold, the sample complexity can become exponentially larger, making dimension transfer statistically impossible without additional structure. Beyond score matching, our framework applies to general invariant $M$-estimators under group actions, yielding dimension-free guarantees for a broad class of statistical and machine learning procedures. These results provide, to our knowledge, the first general theoretical framework for dimension-free scaling laws in invariant score matching, and more broadly for invariant $M$-estimation on variable domains.