We are interested in the two-sample statistical tests to evaluate the hypothesis H₀: {P = Q} against its alternative H₁: {P ≠ Q}. Our data are multivariate, high-dimensional and exhibit strong dependencies between variables. We assume that they are first embedded in a well-chosen RKHS. The construction of our test statistic is inspired by classification problems in machine learning, where the challenge is to find the best separator (linear in the RKHS) between the two groups. The separator is obtained by solving an optimization problem whose solution is a function that defines our test statistic. Consequently, this statistic depends on the optimal discriminant axis between the two groups, particularly involving the within-group covariance operator and the between-groupcovariance operator. To approximate the within-group covariance operator via its empirical estimator, a regularization is required. Unlike the approaches of Harchaoui et al. (2007) or Hagrass et al. (2023) using $L_1$ regularizations, we propose spectral truncation. This method fixes the unknown number $T$ of eigenfunctions to reconstruct the within-group covariance operator. Currently, at a fixed $T$, the test statistic, called truncated kernel Fisher Discriminant Ratio (KFDA\_T), provides a test whose asymptotic calibration is known (Ozier-Lafontaine et al. (2023)).
In this talk, I will present how to theoretically and non-asymptotically bound the p-value of the test associated with the KFDA\_T. The bound is the first step in defining a good calibration of the hyperparameter $T$. In applications, this statistical question is essentiel in the field of genomics, where the two groups are composed of single-cell RNA-seq data. The goal is to detect distinct or similar biological behavior between the groups.

Joint work with Bertrand Michel (Université de Nantes), Franck Picard (ENS de Lyon) et Vincent Rivoirard (Paris-Dauphine).