Finite AMN-Inspired Geometric Regularization for Neural Metric Learning

Neural metric learning is usually evaluated by retrieval accuracy, but a learned dissimilarity can rank examples well while lacking the algebraic structure of a norm. This paper studies Euclidean-anchored residual neural dissimilarities through a finite norm-like geometry inspired by asymptotically metrically normable (AMN) vector spaces. The guiding interpretation is that many learned exponential similarities K = exp(−E/τ ) hide an unbounded energy or distance- like cost behind a bounded affinity. For Laplacian forms the negative logarithm is already distance-like; for Gaussian forms the logarithm first gives a squared distance-like quantity and the square root is AMN-relevant. We formalize this viewpoint by showing that bounded perturbations preserve the large-scale specific energy E(nv, 0)/n, while subadditivity of E becomes multiplicative path consistency of the affinity. The entropy analogy is kept separate: − log K becomes a surprisal term only after normalization of affinities into probabilities. The method penalizes finite violations of homogeneity and subadditivity and couples them with a dyadic estimator of the norm-like component of the learned dissimilarity. The mathematical part formulates residual heads as learned gauges, relates subadditivity to convexity of induced unit balls, constructs a finite norm-envelope witness from sampled latent differences, proves dyadic stability bounds, introduces finite AMN-refinement towers whose witness norms converge under coherence assumptions, and connects kernel-derived distances with quotient latent geometry. The claims are restricted to finite sampled scales in the latent vector space. Full official Fashion-MNIST experiments with supervised-contrastive and proxy-anchor-style Euclidean baselines show that modern Euclidean objectives can outperform the residual family in retrieval, while AMN-inspired terms substantially reduce finite defects inside that family with a small Recall@1 cost. Additional audits compare Euclidean shrinkage, residual flexibility, unseen scales and latent extrapolation. A ten-seed full-query/full-gallery UCI Human Activity Recognition benchmark gives the same qualitative residual-family signal. These results support the method as a finite geometric diagnostic and regularizer for learned latent dissimilarities rather than as a state-of-the-art retrieval objective.