AOBE: Anchored Order-Isomorphic Bi-Lipschitz Embeddings

A representation that faithfully encodes a partially monotonic input order should preserve prescribed comparisons within fixed-context fibers without introducing unsupported comparisons between inputs that are incomparable under that order. Forward partial monotonicity alone does not ensure this: it permits input collapse, comparability between incomparable inputs, and uncontrolled metric distortion. We introduce anchored order-isomorphic bi-Lipschitz embeddings (AOBE), constructed by concatenating a coordinate-wise anchor with bounded positive scales and a norm-controlled partially monotonic network. The anchor retains a positively scaled copy of every direction-aligned constrained coordinate and a signed pair of every context coordinate. For every admissible embedding parameter value, the embedding is an exact order isomorphism from the conditional product order onto its image, is injective, and satisfies parameter-uniform bi-Lipschitz bounds determined by fixed hyperparameters. Across six benchmarks spanning binary classification, regression, and ordinal classification, AOBE achieves competitive predictive performance. In a parameter-matched representation comparison, AOBE is the only evaluated embedding that combines zero audited order-preservation and false-comparability violation rates with positive normalized incomparability margins and a parameter-uniform bi-Lipschitz guarantee. Frozen-channel readouts support task-dependent utility of the nonlinear features, while direct-collapse and task-output audits examine input separation and bounded prediction changes. These results demonstrate that faithful order representation and geometric stability can be enforced while retaining predictive utility.