Finite AMN-Inspired Geometric Regularization for Neural Metric Learning

Neural metric learning is often assessed by retrieval accuracy, but a learned dissimilarity can rank examples well while failing to have norm-like algebraic structure. This paper studies a precise finite question: within a Euclidean-anchored residual family of neural dissimilarities, can one reduce sampled defects of homogeneity, subadditivity, and dyadic reconstruction on latent differences without destroying retrieval performance? The construction is inspired by asymptotically metrically normable (AMN) vector spaces, but its claims are finite, sampled, and latent: it does not prove global AMN rigidity or certify a metric on the input space. The framework is motivated by the observation that many learned similarities have the form K=exp(−E/τ) and therefore encode an unbounded distance-like quantity or squared distance-like quantity behind a bounded affinity. The AMN-relevant object is this cost, not the bounded kernel value. We formalize bounded-perturbation stability of the large-scale specific energy E(nv,0)/n, the conversion of subadditivity into multiplicative affinity consistency, and the quotient interpretation in which directions of zero large-scale cost are collapsed. The mathematical development then introduces finite dyadic diagnostics, learned-gauge and convex-unit-ball interpretations, finite norm-envelope witnesses, dyadic stability bounds, and refinement towers of witness norms. The empirical part reports full official Fashion-MNIST experiments with supervised-contrastive and proxy-anchor-style Euclidean baselines, post hoc audits for shrinkage, residual flexibility, off-training scales, and latent extrapolation, and a ten-seed full-query/full-gallery UCI Human Activity Recognition benchmark. The results show that Euclidean objectives can be stronger for Recall@1, whereas AMN-inspired residual regularization substantially reduces finite norm-like defects inside the residual family. The contribution is therefore a finite diagnostic and regularization framework for learned latent dissimilarities, not a state-of-the-art retrieval objective.