Expressive Power and Approximation Errors of Restricted Boltzmann Machines
This provides theoretical bounds for practitioners using RBMs to ensure model richness and error tolerance, though it is incremental as it builds on existing RBM theory.
The paper tackles the problem of quantifying the expressive power of Restricted Boltzmann Machines (RBMs) by deriving explicit classes of probability distributions they can learn based on the number of units, and it shows that the maximal Kullback-Leibler divergence is bounded by approximately (n - 1) - log(m+1), where n and m are visible and hidden units, respectively.
We present explicit classes of probability distributions that can be learned by Restricted Boltzmann Machines (RBMs) depending on the number of units that they contain, and which are representative for the expressive power of the model. We use this to show that the maximal Kullback-Leibler divergence to the RBM model with $n$ visible and $m$ hidden units is bounded from above by $n - \left\lfloor \log(m+1) \right\rfloor - \frac{m+1}{2^{\left\lfloor\log(m+1)\right\rfloor}} \approx (n -1) - \log(m+1)$. In this way we can specify the number of hidden units that guarantees a sufficiently rich model containing different classes of distributions and respecting a given error tolerance.