  This paper proposes a multi-grid method for learning energy-based generative
ConvNet models of images. For each grid, we learn an energy-based probabilistic
model where the energy function is defined by a bottom-up convolutional neural
network (ConvNet or CNN). Learning such a model requires generating synthesized
examples from the model. Within each iteration of our learning algorithm, for
each observed training image, we generate synthesized images at multiple grids
by initializing the finite-step MCMC sampling from a minimal 1 x 1 version of
the training image. The synthesized image at each subsequent grid is obtained
by a finite-step MCMC initialized from the synthesized image generated at the
previous coarser grid. After obtaining the synthesized examples, the parameters
of the models at multiple grids are updated separately and simultaneously based
on the differences between synthesized and observed examples. We show that this
multi-grid method can learn realistic energy-based generative ConvNet models,
and it outperforms the original contrastive divergence (CD) and persistent CD.
