Core Concepts
Permutation invariance simplifies complex problems in machine learning, with statistical tests and dimension reduction playing crucial roles.
Abstract
Permutation invariance is essential in various applications like anomaly detection and text classification. Research focuses on testing permutation invariance of multivariate distributions and reducing dimensions through statistical methods. The sorting trick simplifies the exploitation of permutation invariance, leading to more efficient models. Kernel ridge regression of permutation invariant functions is a simple yet effective approach for nonlinear function recovery. Theoretical guarantees support the performance of these methods, showcasing their potential for practical applications.
Stats
T := supt∈[0,1]d √n eFn(t) − Fn(t)
W := sup t∈[0,1]d 1 √n n X i=1 ([ti ≤ sort t] − [ti ≤ t]) ei
T := supt∈[0,1]d √n eFn(t) − Fn(t)
W := sup t∈[0,1]d 1 √n n X i=1 ([ti ≤ sort t] − [ti ≤ t]) ei
Cov: 0.96, 0.99, < 0.01, < 0.01, 0.09, 0.29
Pow: 0.04, 0.01, > 0.99, > 0.99, 0.91, 0.71
Cov: 0.95, > 0.99...
Pow: ...
Bias and variance comparisons with n = 10000...