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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.
Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid with a given grid point density . Assume furthermore that a solution on any grid N i {\displaystyle N_{i}} may be obtained with a given effort W i = ρ K N i {\displaystyle W_{i}=\rho KN_{i}} from a solution on a coarser grid i + 1 ...
Nonlinear PCA (NLPCA) uses backpropagation to train a multi-layer perceptron (MLP) to fit to a manifold. [37] Unlike typical MLP training, which only updates the weights, NLPCA updates both the weights and the inputs. That is, both the weights and inputs are treated as latent values.
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 ...
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 ...
Below is an example of a learning algorithm for a single-layer perceptron with a single output unit. For a single-layer perceptron with multiple output units, since the weights of one output unit are completely separate from all the others', the same algorithm can be run for each output unit.
This equation represents the underlying geometry of the chaotic time series generated by the logistic map. Generation of the time series from this equation is the forward problem. The examples here illustrate the inverse problem; identification of the underlying dynamics, or fundamental equation, of the logistic map from exemplars of the time ...
For example, multilayer perceptron (MLPs) and time delay neural network (TDNNs) have limitations on the input data flexibility, as they require their input data to be fixed. Standard recurrent neural network (RNNs) also have restrictions as the future input information cannot be reached from the current state.