### Stochastic Feedforward Neural Networks: Universal Approximation

In this chapter we take a look at the universal approximation question for stochastic feedforward neural networks. In contrast to deterministic networks, which represent mappings from a set of inputs to a set of outputs, stochastic networks represent mappings from a set of inputs to a set of probability distributions over the set of outputs. In particular, even if the sets of inputs and outputs are finite, the class of stochastic mappings in question is not finite. Moreover, while for a deterministic function the values of all output variables can be computed independently of each other given the values of the inputs, in the stochastic setting the values of the output variables may need to be correlated, which requires that their values are computed jointly. A prominent class of stochastic feedforward networks which has played a key role in the resurgence of deep learning are deep belief networks. The representational power of these networks has been studied mainly in the generative setting, as models of probability distributions without an input, or in the discriminative setting for the special case of deterministic mappings. We study the representational power of deep sigmoid belief networks in terms of compositions of linear transformations of probability distributions, Markov kernels, that can be expressed by the layers of the network. We investigate different types of shallow and deep architectures, and the minimal number of layers and units per layer that are sufficient and necessary in order for the network to be able to approximate any given stochastic mapping from the set of inputs to the set of outputs arbitrarily well.

### Deep Narrow Boltzmann Machines are Universal Approximators

We show that deep narrow Boltzmann machines are universal approximators of probability distributions on the activities of their visible units, provided they have sufficiently many hidden layers, each containing the same number of units as the visible layer. We show that, within certain parameter domains, deep Boltzmann machines can be studied as feedforward networks. We provide upper and lower bounds on the sufficient depth and width of universal approximators. These results settle various intuitions regarding undirected networks and, in particular, they show that deep narrow Boltzmann machines are at least as compact universal approximators as narrow sigmoid belief networks and restricted Boltzmann machines, with respect to the currently available bounds for those models.

### Geometry and Expressive Power of Conditional Restricted Boltzmann Machines

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results their ability to represent conditional Markov random fields and conditional distributions with restricted supports, the minimal size of universal approximators, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional probability models, which allow us to improve the results that can be derived from existing work on restricted Boltzmann machine probability models.

### On the Number of Linear Regions of Deep Neural Networks

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers.

### Universal Approximation Depth and Errors of Narrow Belief Networks with Discrete Units

We generalize recent theoretical work on the minimal number of layers of narrow deep belief networks that can approximate any probability distribution on the states of their visible units arbitrarily well. We relax the setting of binary units (Sutskever and Hinton, 2008; Le Roux and Bengio, 2008, 2010; Mont\'ufar and Ay, 2011) to units with arbitrary finite state spaces, and the vanishing approximation error to an arbitrary approximation error tolerance. For example, we show that a $q$-ary deep belief network with $L\geq 2+\frac{q^{\lceil m-\delta \rceil}-1}{q-1}$ layers of width $n \leq m + \log_q(m) + 1$ for some $m\in \mathbb{N}$ can approximate any probability distribution on $\{0,1,\ldots,q-1\}^n$ without exceeding a Kullback-Leibler divergence of $\delta$. Our analysis covers discrete restricted Boltzmann machines and na\"ive Bayes models as special cases.