  Many parametric statistical models are not properly normalised and only
specified up to an intractable partition function, which renders parameter
estimation difficult. Examples of unnormalised models are Gibbs distributions,
Markov random fields, and neural network models in unsupervised deep learning.
In previous work, the estimation principle called noise-contrastive estimation
(NCE) was introduced where unnormalised models are estimated by learning to
distinguish between data and auxiliary noise. An open question is how to best
choose the auxiliary noise distribution. We here propose a new method that
addresses this issue. The proposed method shares with NCE the idea of
formulating density estimation as a supervised learning problem but in contrast
to NCE, the proposed method leverages the observed data when generating noise
samples. The noise can thus be generated in a semi-automated manner. We first
present the underlying theory of the new method, show that score matching
emerges as a limiting case, validate the method on continuous and discrete
valued synthetic data, and show that we can expect an improved performance
compared to NCE when the data lie in a lower-dimensional manifold. Then we
demonstrate its applicability in unsupervised deep learning by estimating a
four-layer neural image model.
