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If a multilayer perceptron has a linear activation function in all neurons, that is, a linear function that maps the weighted inputs to the output of each neuron, then linear algebra shows that any number of layers can be reduced to a two-layer input-output model.
A neuron with label receiving an input () from predecessor neurons consists of the following components: [1] an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter,
In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural ...
When multiple layers use the identity activation function, the entire network is equivalent to a single-layer model. Range When the range of the activation function is finite, gradient-based training methods tend to be more stable, because pattern presentations significantly affect only limited weights.
Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input–output example, and does so efficiently, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this can be derived through ...
In particular, this shows that a perceptron network with a single infinitely wide hidden layer can approximate arbitrary functions. Such an f {\displaystyle f} can also be approximated by a network of greater depth by using the same construction for the first layer and approximating the identity function with later layers.
Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} .
A two-layer neural network capable of calculating XOR. The numbers within the neurons represent each neuron's explicit threshold. The numbers that annotate arrows represent the weight of the inputs. Note that If the threshold of 2 is met then a value of 1 is used for the weight multiplication to the next layer.