  Vanishing component analysis (VCA) computes approximate generators of
vanishing ideals of samples, which are further used for extracting nonlinear
features of the samples. Recent studies have shown that normalization of
approximate generators plays an important role and different normalization
leads to generators of different properties. In this paper, inspired by recent
self-supervised frameworks, we propose a contrastive normalization method for
VCA, where we impose the generators to vanish on the target samples and to be
normalized on the transformed samples. We theoretically show that a contrastive
normalization enhances the discriminative power of VCA, and provide the
algebraic interpretation of VCA under our normalization. Numerical experiments
demonstrate the effectiveness of our method. This is the first study to tailor
the normalization of approximate generators of vanishing ideals to obtain
discriminative features.
