Universal representation by Boltzmann machines with Regularised Axons
This work addresses the training and sampling bottlenecks in Boltzmann machines for machine learning practitioners, though it is incremental as it builds on prior regularization proposals.
The authors tackled the problem of Boltzmann machines being hard to sample and train due to complex energy landscapes, and proved that regularizing connections preserves their ability to represent arbitrary distributions while enabling efficient guided sampling and training, with explicit results showing exponential storage of correlated patterns.
It is widely known that Boltzmann machines are capable of representing arbitrary probability distributions over the values of their visible neurons, given enough hidden ones. However, sampling -- and thus training -- these models can be numerically hard. Recently we proposed a regularisation of the connections of Boltzmann machines, in order to control the energy landscape of the model, paving a way for efficient sampling and training. Here we formally prove that such regularised Boltzmann machines preserve the ability to represent arbitrary distributions. This is in conjunction with controlling the number of energy local minima, thus enabling easy \emph{guided} sampling and training. Furthermore, we explicitly show that regularised Boltzmann machines can store exponentially many arbitrarily correlated visible patterns with perfect retrieval, and we connect them to the Dense Associative Memory networks.