# Deep Neural Networks (DNNs)

In this video, Dr Alex Tse introduces Deep neural networks (DNNs), which is a generalization of shallow neural networks.

In Step 3.4, we introduce the shallow neural network. In this video, Dr Alex Tse introduces Deep neural networks (DNNs). DNNs can be regarded as the extension of the shallow neural network, as it allows multiple hidden layers in contrast to the single hidden layer of the shallow neuron network.

We summarize the architecture of DNNs as follows:

Let (h^{(l)}) denote the (l^{th}) layer with (n_{l}) neurons.

• Input layer ((h^{(0)}: mathbb{R}^{d} rightarrow mathbb{R}^{n_{0}}) with (n_0 = d)):

(h^{(0)}(x) = x, forall x in mathbb{R}^d).

• Hidden layer ((h^{(l)}: mathbb{R}^{d} rightarrow mathbb{R}^{n_{l}})): (forall l in {1, 2, cdots, L-1}).
• Output layer ((h^{(L)}: mathbb{R}^{d} rightarrow mathbb{R}^{e})).

((h^{(l)})_{l = 1}^{L}) is defined in the following recursive way that for any (x in mathbb{R}^{d}),

[z^{(l+1)}(x) = h^{(l)}(x) W^{(l)} + b^{(l)},] [h^{(l+1)}(x) = sigma_{l+1}(z^{(l+1)}(x)),]

where (W^{(l)}) is a (n_{l} times n_{l+1}) matrix, (b^{(l)}) is a (n_{l+1}) dimensional row vector and (sigma_{l}) is the activation function on the (l^{th}) layer. Here (theta:= (W^{(l)}, b^{(l)})_{l=1}^{L-1}) is the parameters of DNNs, which can be trained from data.