🤖 AI Summary
This work addresses the limitation of traditional quantile regression in modeling pairwise similarity responses—such as image similarity scores in face recognition—by introducing, for the first time, a pairwise quantile regression framework. The proposed method predicts conditional quantiles of similarity scores for input pairs by minimizing a pairwise pinball loss. On the theoretical front, sharp concentration inequalities based on U-processes are employed to establish generalization error bounds, yielding fast learning rates under mild assumptions. Empirically, the approach is validated through both simulated data and real-world face recognition similarity scoring tasks, demonstrating its effectiveness and practical utility.
📝 Abstract
Quantile regression provides a powerful tool for summarizing the conditional distribution of a real valued random variable (r.v.) of interest $Y$ as a function of covariates $Z$ in cases where it shows a large dispersion with high probability, going beyond the situation where standard least square regression is informative/predictive. This article aims to extend this methodology to the pairwise case, when the variable to be explained takes the form of a similarity function between two independent observations, such as pixelated ID photos, as input data of biometric systems) and the explanatory variables take the form of a pair of covariates of the observations, such as the age or the hair color. We establish theoretical guarantees for solutions of this statistical learning problem, considered here as empirical minimizers of a pairwise version of the pinball loss. Leveraging sharp concentration results for $U$-processes, we prove generalization bounds and identify mild conditions under which fast learning rates can be achieved. Confirming the probabilistic analysis, experiments based on simulation data also provide solid empirical evidence of the validity of the methodology promoted here for pairwise quantile regression. Finally, its usefulness from an application perspective is demonstrated by a detailed study aimed at analyzing errors in similarity scoring for facial recognition.