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.

通过对玻尔兹曼机的连接进行正则化，我们证明了正则化玻尔兹曼机能够表示任意分布，并且控制能量局部极小值的数量，从而实现了简化的采样和训练，同时能够存储指数级的任意相关可见模式，并与稠密关联记忆网络相关联。