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In this survey we discuss various approximation-theoretic problems that arise in the multilayer feedforward perceptron (MLP) model in neural networks. The MLP model is one of the more popular and practical of the many neural network models. Mathematically it is also one of the simpler models. Nonetheless the mathematics of this model is not well understood, and many of these problems are approximation-theoretic in character. Most of the research we will discuss is of very recent vintage. We will report on what has been done and on various unanswered questions. We will not be presenting practical (algorithmic) methods. We will, however, be exploring the capabilities and limitations of this model.

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Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given.

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