Estimating Densities with Non-Parametric Exponential Families

We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-parametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential family. Using the proposed approach, we modify the exponential random graph model, commonly used for modeling small-size graph distributions, to address the well-known issue of model degeneracy.Comment: 22 pages, 5 figure

Generalization and Memorization: The Bias Potential Model

Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or diverges.Comment: Added new section on regularized mode

Estimating Densities with Non-Parametric Exponential Families

We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a nonparametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential family. Using the proposed approach, we modify the exponential random graph model, commonly used for modeling small-size graph distributions, The problem of density estimation is ubiquitous in machine learning and statistics. A typical approach would assume a parametric family for the distribution from which the observed data is drawn and estimate the parameters by fitting them to the data. Among the parametric families, exponential families play a prominent role, as maximum likelihood estimation from complete data for exponential families is asymptoticall