Conditional Noise-Contrastive Estimation of Unnormalised Models

TL;DR

This paper introduces a new semi-automated noise-contrastive estimation method for unnormalised models, improving upon traditional NCE by leveraging observed data to generate noise, with theoretical validation and practical applications in deep learning.

Contribution

The paper proposes a novel noise-contrastive estimation approach that uses observed data for noise generation, bridging NCE and score matching, and demonstrating improved performance in deep learning models.

Findings

The new method generalizes NCE and score matching.

It performs better on data in lower-dimensional manifolds.

Validated on synthetic data and neural image models.

Abstract

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.