Deep Boltzmann Machines
Ruslan Salakhutdinov, Geoffrey Hinton; JMLR W&CP 5:448-455, 2009.
We present a new learning algorithm for Boltzmann machines that contain many layers of hidden variables. Data-dependent expectations are estimated using a variational approximation that tends to focus on a single mode, and data-independent expectations are approximated using persistent Markov chains. The use of two quite different techniques for estimating the two types of expectation that enter into the gradient of the log-likelihood makes it practical to learn Boltzmann machines with multiple hidden layers and millions of parameters. The learning can be made more efficient by using a layer-by-layer ``pre-training'' phase that allows variational inference to be initialized by a single bottom-up pass. We present results on the MNIST and NORB datasets showing that deep Boltzmann machines learn good generative models and perform well on handwritten digit and visual object recognition tasks.