A Linear-Time Kernel Goodness-of-Fit Test

Wittawat Jitkrittum (1) , Wenkai Xu (1) , Zoltán Szabó (2) , Kenji Fukumizu (3) , Arthur Gretton (1)

1 Gatsby Computational Neuroscience Unit
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 Tokyo - The Institute of Statistical Mathematics

We propose a novel adaptive test of goodness-of-fit, with computational cost linear in the number of samples. We learn the test features that best indicate the differences between observed samples and a reference model, by minimizing the false negative rate. These features are constructed via Stein’s method, meaning that it is not necessary to compute the normalising constant of the model. We analyse the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift alternative, our test always has greater relative efficiency than a previous linear-time kernel test, regardless of the choice of parameters for that test. In experiments, the performance of our method exceeds that of the earlier linear-time test, and matches or exceeds the power of a quadratic-time kernel test. In high dimensions and where model structure may be exploited, our goodness of fit test performs far better than a quadratic-time two-sample test based on the Maximum Mean Discrepancy, with samples drawn from the model.

Domaines: Statistiques [math.ST]; Autres [stat.ML]; Apprentissage [cs.LG]

Wittawat Jitkrittum, Wenkai Xu, Zoltán Szabó, Kenji Fukumizu, Arthur Gretton. A Linear-Time Kernel Goodness-of-Fit Test. [Research Report] University College London; École Polytechnique; The Institute of Statistical Mathematics. 2017. ⟨hal-01527717v1⟩