The mean field algorithm is a widely used approximate inference algorithm for graphical models whose exact inference is intractable. In each iteration of mean field, the approximate marginals for each variable are updated by getting information from the neighbors. This process can be equivalently converted into a feedforward network, with each layer representing one iteration of mean field and with tied weights on all layers. This conversion enables a few natural extensions, e.g. untying the weights in the network. In this paper, we study these mean field networks (MFNs), and use them as inference tools as well as discriminative models. Preliminary experiment results show that MFNs can learn to do inference very efficiently and perform significantly better than mean field as discriminative models.
Another work by @cite_3 briefly draws a connection between mean field inference of a specific binary MRF with neural networks, but did not explore further variations.
@cite_3
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Markov Random Fields (MRFs) are a widely used graphical model, but the inference problem is NP-hard. For first-order MRFs with binary labels, Dead End Elimination (DEE) [7] and QPBO [2, 14] can find the optimal labeling for some variables, the much harder case of larger label sets has been addressed by Kovtun [16, 17] and related methods [12, 23, 24, 25], which impose substantial computational overhead. We describe an efficient algorithm to correctly label a subset of the variables for arbitrary MRFs, with particularly good performance on binary MRFs. We propose a sufficient condition to check if a partial labeling is optimal, which is a generalization of DEE's purely local test. We give a hierarchy of relaxations that provide larger optimal partial labelings at the cost of additional computation. Empirical studies were conducted on several benchmarks, using expansion moves [4] for inference. Our algorithm runs in a few seconds, and improves the speed of MRF inference with expansion moves by a factor of 1.5 to 12. Until recently, the lack of ground truth data has hindered the application of discriminative structured prediction techniques to the stereo problem. In this paper we use ground truth data sets that we have recently constructed to explore different model structures and parameter learning techniques. To estimate parameters in Markov random fields (MRFs) via maximum likelihood one usually needs to perform approximate probabilistic inference. Conditional random fields (CRFs) are discriminative versions of traditional MRFs. We explore a number of novel CRF model structures including a CRF for stereo matching with an explicit occlusion model. CRFs require expensive inference steps for each iteration of optimization and inference is particularly slow when there are many discrete states. We explore belief propagation, variational message passing and graph cuts as inference methods during learning and compare with learning via pseudolikelihood. To accelerate approximate inference we have developed a new method called sparse variational message passing which can reduce inference time by an order of magnitude with negligible loss in quality. Learning using sparse variational message passing improves upon previous approaches using graph cuts and allows efficient learning over large data sets when energy functions violate the constraints imposed by graph cuts. Many computer vision problems can be formulated in a Bayesian framework with Markov Random Field (MRF) or Conditional Random Field (CRF) priors. Usually, the model assumes that a full Maximum A Posteriori (MAP) estimation will be performed for inference, which can be really slow in practice. In this paper, we argue that through appropriate training, a MRF CRF model can be trained to perform very well on a suboptimal inference algorithm. The model is trained together with a fast inference algorithm through an optimization of a loss function on a training set containing pairs of input images and desired outputs. A validation set can be used in this approach to estimate the generalization performance of the trained system. We apply the proposed method to an image denoising application, training a Fields of Experts MRF together with a 1-4 iteration gradient descent inference algorithm. Experimental validation on unseen data shows that the proposed training approach obtains an improved benchmark performance as well as a 1000-3000 times speedup compared to the Fields of Experts MRF trained with contrastive divergence. Using the new approach, image denoising can be performed in real-time, at 8 fps on a single CPU for a 256 × 256 image sequence, with close to state-of-the-art accuracy. We present an approach to low-level vision that combines two main ideas: the use of convolutional networks as an image processing architecture and an unsupervised learning procedure that synthesizes training samples from specific noise models. We demonstrate this approach on the challenging problem of natural image denoising. Using a test set with a hundred natural images, we find that convolutional networks provide comparable and in some cases superior performance to state of the art wavelet and Markov random field (MRF) methods. Moreover, we find that a convolutional network offers similar performance in the blind de-noising setting as compared to other techniques in the non-blind setting. We also show how convolutional networks are mathematically related to MRF approaches by presenting a mean field theory for an MRF specially designed for image denoising. Although these approaches are related, convolutional networks avoid computational difficulties in MRF approaches that arise from probabilistic learning and inference. This makes it possible to learn image processing architectures that have a high degree of representational power (we train models with over 15,000 parameters), but whose computational expense is significantly less than that associated with inference in MRF approaches with even hundreds of parameters.