This simple modification brings increased representational power, as illustrated by two multimodal examples of the rectified Gaussian, the competitive and the cooperative distributions. The modes of the competitive distribution are well-separated by regions of low probability. The modes of the cooperative distribution are closely spaced along a nonlinear continuous manifold. Neither distribution can be accurately approximated by a single standard Gaussian. In short, the rectified Gaussian is able to represent both discrete and continuous variability in a way that a standard Gaussian cannot.

This simple modification brings increased representational power, as illustrated by two multimodal examples of the rectified Gaussian, the competitive and the cooperative distributions. The modes of the competitive distribution are well-separated by regions of low probability. The modes of the cooperative distribution are closely spaced along a nonlinear continuous manifold. Neither distribution can be accurately approximated by a single standard Gaussian. In short, the rectified Gaussian is able to represent both discrete and continuous variability in a way that a standard Gaussian cannot.

The variables of the rectified Gaussian are constrained to be nonnegative, enabling the use of nonconvex energy functions.Two multimodal examples, the competitive and cooperative distributions, illustrate the representational power of the rectified Gaussian. Since the cooperative distribution can represent thetranslations of a pattern, it demonstrates the potential of the rectified Gaussian for modeling pattern manifolds.