Scalable and Interpretable Representation Alignment with Ordinal Similarity

A new framework for measuring how similar two machine-learning representations are promises to solve long-standing problems with existing metrics, according to a paper submitted on 15 June 2026. The ordinal-similarity approach is designed to be interpretable, robust to outliers, and computationally scalable for large datasets. [1] The framework, detailed on arXiv, is instantiated by two indices: the Triplet Similarity Index (TSI) and the Quadruplet Similarity Index (QSI). Both measure alignment by quantifying the consistency of ordinal relationships—essentially, whether the relative ordering of distances between data points is preserved across different representations. [1] [2] Existing metrics for representation similarity, such as Centered Kernel Alignment (CKA), have significant drawbacks. The authors note that these metrics often lack interpretability because their baselines shift depending on the data, are vulnerable to outliers, and become computationally intractable on large datasets, forcing practitioners to rely on heuristic approximations. [2] [3] In contrast, the ordinal-similarity framework provides a direct probabilistic interpretation and establishes stable baselines for dissimilar representations, a property formally proven in the paper. The researchers also demonstrate a formal equivalence between TSI and local neighborhood alignment, as measured by Mutual Nearest Neighbors. [2] [4] On the computational side, the paper provides efficient algorithms for the exact calculation of the indices and introduces a Monte Carlo sampling scheme with provable approximation guarantees. These guarantees are independent of dataset size, enabling high-confidence evaluation on massive datasets such as ImageNet. [3] [5] Empirical validation tracked the training dynamics of Vision Transformers and alignment patterns in multimodal CLIP models. The results showed that TSI and QSI correctly tracked alignment across varying model scales and training regimes, even as embedding dimensions changed—conditions where metrics like CKA often fail. The proposed sampling schemes achieved approximation precision described as orders of magnitude higher than standard batching. [3] [4] The work addresses a fundamental challenge in representation learning, a subfield of machine learning where algorithms build models from training data to make predictions. [7] By prioritizing the ordering of distances over their raw values, the ordinal-similarity framework offers a principled method for comparing the internal data geometries that different models learn. [1] [2]