Due to the similarities between a Neural Autoregressive Density Estimator (NADE) and a Restricted Boltzmann Machine (RBM), one can perform mean field inferences on the NADE system in a computationally similar way. However, as the RBM is a recurrent network, only one inference can be made at a time. This is in contrast to the discrete nature of a NADE network, which allows it to perform a finite number of inferences $$k$$ simultaneously, where $$k$$ is less than the total number of inputs the NADE network was trained to handle.

University of Toledo

Google

-	Restricted Boltzmann Machine is an undirected graphical model that plays a major role in Deep Learning Framework in recent times. It was initially introduced as Harmonium by Paul Smolensky in 1986 and it gained big popularity in recent years in the context of the Netflix Prize where Restricted Boltzmann Machines achieved state of the art performance in collaborative filtering and have beaten most of the competition.


Restricted Boltzmann Machine

The mean field approach is a commonly used distribution set that assumes $$q$$ is of the following form: $$q(h|v)=\prod_i q(h_i|v)$$ This specification for the factors of $$q$$ allows $$q(h|v)$$ to be optimized by optimizing a finite number of lower dimensional functions $$q(h_i|v)$$ using traditional optimization techniques. This is only one of many ways that the distribution $$q$$ can be limited. The technique is referred to as "structured variational inference" in general.

Mean Field Approach in Variational Inference

A Neural Autoregressive Density Estimator (NADE) is largely identical to a standard Neural Auto-Regressive Network, but with one key difference in parameter sharing. Input values $$x_i$$ are supplied into the network only after being weighted by weight vectors $$W'_{j,k,i} = W_{k,i}$$. In other words, the weight vector $$W_{k,i}$$ is reused for all inputs into hidden unit groups $$h^{(j)}$$ where $$j \geq i$$. This allows the forward propagation of a NADE model to resemble the computation of missing inputs in a mean field inference for a Restricted Boltzmann Machine (RBM).

Neural Autoregressive Density Estimator (NADE)

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

- On its forward pass, an RBM uses inputs to make predictions about node activations.
- On its backward pass, when activations are fed in and reconstructions, or guesses about the original data, are spit out, an RBM is attempting to estimate the probability of inputs given activations of hidden layer, which are weighted with the same coefficients as those used on the forward pass. 
- Reconstruction is making guesses about the probability distribution of the original input; i.e. the values of many varied points at once. This is known as generative learning.
- To measure the distance between its estimated probability distribution and the ground-truth distribution of the input, RBMs use Kullback Leibler Divergence. 
- KL-Divergence measures the non-overlapping, or diverging, areas under the two curves, and an RBM’s optimization algorithm attempts to minimize those areas so that the shared weights, when multiplied by activations of hidden layer one, produce a close approximation of the original input. 

Probability Distribution and Generative Learning

- Contrastive Divergence (CD)
- Stochastic Maximum Likelihood (SML)
- Persistent Contrastive Divergence (PCD)
- Ratio matching

Training Models for RBM

Replacing matrix multiplication by discrete convolution with a small kernel is the standard way to solve these problems for inputs with translation-invariant spatial or temporal structure. This convolution approach works well when applied to RBMs. Lee et al. demonstrated that probabilistic max-pooling could be used to build convolutional deep Boltzmann machines. 


Convolutional Boltzmann Machines

It is an algorithm which is useful for dimensionality reduction, classification, regression, collaborative filtering, feature learning, and topic modeling.

Restricted Boltzmann Machines are shallow, two-layer neural nets that constitute the building blocks of deep-belief networks. The first layer of the RBM is called the visible, or input layer, and the second is the hidden layer.

The restriction in a Restricted Boltzmann Machine is that there is no intra-layer communication.

Restricted Boltzmann Machine Applications

NADE-k

The canonical Restricted Boltzmann Machine (RBM) is an energy-based model with binary visible and hidden units. Its energy function is:

$$E(v,h) = -b^Tv - c^Th - v^TWh$$

where $$b$$, $$c$$, and $$W$$ are unconstrained, real-valued, learnable parameters. It is easy to take its derivatives:

$$\frac{\partial}{\partial W_{i,j}}E(v,h) = -v_ih_j$$

There are no direct interactions between any two visible units or any two hidden units (restrictions). The restrictions on the RBM structure yield the properties:

$$p(h|v)=\prod_ip(h_i|v)$$
$$p(v|h) = \prod_ip(v_i|h)$$

The individual conditionals are simple to compute. For the binary RBM we obtain:

$$P(h_i=1|v) = \rho(v^TW_{:,i} + b_i)$$
$$P(h_i=0|v) = 1 - \rho(v^TW_{:,i} + b_i)$$

Together, these properties allow for efficient block Gibbs sampling and efficient derivatives, making training convenient. Samples generated by Gibbs sampling from an RBM model are shown below:

Mathematical Formulation of Restricted Boltzmann Machines

Restricted Boltzmann Machines (RBMs) are shallow, two-layer neural networks that constitute the building blocks of deep-belief networks. The first layer of the RBM is the visible (input) layer, and the second is the hidden layer. Because inputs from all visible nodes are passed to all hidden nodes, an RBM can be defined as a symmetrical bipartite graph.

During the forward pass, each node of the visible layer is multiplied by a separate weight, the products are summed, added to a bias, and the result is passed through an activation function to produce the hidden layer's output. In the reconstruction phase, the hidden layer's activations become the input in a backward pass. They are multiplied by the same weights, one per internode edge. The sum of those products is added to a visible-layer bias at each visible node, producing a reconstruction (an approximation of the original input). This can be represented by the following diagram:

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