Global Average Precision for Representation Learning

What can a distribution-matching regularizer such as SIGReg in LeJEPA certify about contrastive learning? We study shared Gaussianization (SG), a characteristic-function Gaussianity test on the average of two normalized views, scaled by an independent $χ_d$ radius. Because disagreeing views shorten the average, one test detects both misalignment and non-uniformity. SG vanishes exactly at the aligned, uniform minimizers of population InfoNCE, and under equal marginals it bounds the InfoNCE excess by $4\cdot 3^{3/4}β$ times the square root of the SG loss, plus a term linear in the loss. The square-root rate and this dimension-free constant are sharp, and no squared mean-embedding distance on view pairs achieves a faster rate. With an explicit alignment term, a rotation-invariant uniformity test gives a linear bound if and only if its spectrum dominates that of InfoNCE's kernel $e^{βu^\top v}$; SG's own test does, Gaussian kernels $e^{-γ\|u-v\|^2}$ qualify exactly when $γ\ge β/2$, and moment matching never does. Away from the optimum, the objectives differ. Along an isotropic nuisance channel, pure SG lowers its loss by adding per-view nuisance whenever the shared code is non-uniform. An alignment weight above the channel's gain makes the nuisance-free solution a strict local minimizer; for LeJEPA, the same rule gives a critical SIGReg weight that decreases with the batch size. At finite batch size, an off-diagonal U-statistic removes a plug-in bias toward misalignment. In controlled latent-variable models, pure SG retains per-view style, an alignment weight above the measured gain removes it, and for LeJEPA at three batch sizes the measured gain separates the encoders that retain style from those that do not. InfoNCE training also reaches a lower SG$_{0.2}$ loss than SG$_{0.2}$ training from scratch, which points to an optimization gap.

Evaluating representation similarity is fundamental to representation learning. However, existing metrics suffer from significant limitations: they lack interpretability due to shifting baselines, lack robustness to outliers, and are computationally intractable for large datasets, forcing reliance on heuristic approximations. To address this, we develop an ordinal-similarity framework, instantiated by the Triplet (TSI) and Quadruplet (QSI) Similarity Indices, which measure alignment by quantifying the consistency of ordinal relationships. We theoretically demonstrate this formulation is inherently interpretable, robust to outliers, and computationally efficient. Finally, we establish a formal equivalence between TSI and local neighborhood alignment, measured by Mutual Nearest Neighbors. Empirically, we validate these properties and show that ordinal similarity offers a scalable approach to measuring alignment, enabling practitioners to better understand and design representations.

Approximate nearest neighbor (ANN) search has become a core primitive in information retrieval and modern machine learning tasks, from classification to retrieval-augmented generation. The community evaluates and tunes ANN algorithms primarily on their throughput at a given Recall@k, the fraction of true exact neighbors retrieved. We argue that what really matters in ANN search is the quality of the retrieved results and not their overlap with the true kNN set. We show that using Recall@k to assess retrieval quality forces unnecessary computational overhead and investigate replacing it by 1/Ratio@k, the inverse approximation ratio. 1/Ratio@k evaluates the differences between the distances of the retrieved and true neighbors. It is judge-free, hyperparameter-free, and computable from standard ANN benchmark inputs alone. We benchmark state-of-the-art ANN algorithms across diverse datasets spanning a wide range of intrinsic dimensionalities, evaluating the two metrics comprehensively across efficiency, downstream classification, and retrieval-augmented generation. On the efficiency axis, optimizing for 1/Ratio@k reaches operational quality thresholds at a substantially lower computational cost than Recall@k. In downstream tasks, performance indicators (label precision, semantic similarity, BERTScore, and LLM-graded quality) remain highly stable even when Recall@k drops significantly. The inverse approximation ratio, on the other hand, closely mirrors this stability, tracking true utility much better than Recall@k. Ultimately, while Recall@k overstates the true cost of approximation, 1/Ratio@k offers a more accurate, deployable proxy for actual ANN quality.