We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multi-category classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this auto-encoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns.
It is well known that natural images admit sparse representations by redundant dictionaries of basis functions such as Gabor-like wavelets. However, it is still an open question as to what the next layer of representational units above the layer of wavelets should be. We address this fundamental question by proposing a sparse FRAME (Filters, Random field, And Maximum Entropy) model for representing natural image patterns. Our sparse FRAME model is an inhomogeneous generalization of the original FRAME model. It is a non-stationary Markov random field model that reproduces the observed statistical properties of filter responses at a subset of selected locations, scales and orientations. Each sparse FRAME model is intended to represent an object pattern and can be considered a deformable template. The sparse FRAME model can be written as a shared sparse coding model, which motivates us to propose a two-stage algorithm for learning the model. The first stage selects the subset of wavelets from the dictionary by a shared matching pursuit algorithm. The second stage then estimates the parameters of the model given the selected wavelets. Our experiments show that the sparse FRAME models are capable of representing a wide variety of object patterns in natural images and that the learned models are useful for object classification.