Quantitative Universal Approximation Bounds for Deep Belief Networks
This provides foundational theoretical guarantees for deep belief networks in density estimation, which is incremental but important for machine learning theory.
The paper tackles the problem of approximating multivariate probability densities with deep belief networks, showing they can approximate any density under mild integrability conditions, with sharp quantitative bounds on the error in terms of hidden units.
We show that deep belief networks with binary hidden units can approximate any multivariate probability density under very mild integrability requirements on the parental density of the visible nodes. The approximation is measured in the $L^q$-norm for $q\in[1,\infty]$ ($q=\infty$ corresponding to the supremum norm) and in Kullback-Leibler divergence. Furthermore, we establish sharp quantitative bounds on the approximation error in terms of the number of hidden units.