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- Machine learning
Two-layers neural network
Mathematically, a simple two-layers neural network with relu non-linearities can be written like below. For an input vector \(x \in \mathbb{R}^D\), \(\mathbf{a} = (a_1, \cdots, a_N)\in \mathbb{R}^M\) are the output weights, \(\mathbf{b} = (b_1, \cdots, b_N)\in \mathbb{R}^D\) are the input weights
\[ h(x) = \frac{1}{m} \sum_{i=1}^m a_i \max\{ b_i^\top x,0\}, \]
Universal approximation theorem
Cybenko showed in 1989 that a neural network of arbitrary width with sigmoid activation function could approximate any continuous function (Cybenko 1989).
Barron added rates of convergence by enforcing smoothness condition on the target function (Barron 1993).
Bibliography
- Barron, A. R.. May 1993. "Universal Approximation Bounds for Superpositions of a Sigmoidal Function". IEEE Transactions on Information Theory 39 (3):930–45.
- Cybenko, G.. December 1989. “Approximation by Superpositions of a Sigmoidal Function”. Mathematics of Control, Signals, and Systems 2 (4):303–14.