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Most brains have lots of neurons, each neuron approximates a linear-threshold unit.
Perceptrons (one-layer networks) approximate neurons, but are as such insufficiently expressive.
Multi-layer networks are sufficiently expressive; can be trained to deal with generalized data sets, i.e. via error back-propagation.
Multi-layer networks allow for the modeling of arbitrary separation boundaries, while single-layer perceptrons only provide linear boundaries.
Endless number of applications: Handwriting Recognition, Time Series Prediction, Bioinformatics, Kernel Machines (Support Vectore Machines), Data Compression, Financial Predication, Speech Recognition, Computer Vision, Protein Structures...