1. M. J. Donahue, L. Gurvits, C. Darken, and E.D. Sontag. Rates of convex approximation in non-Hilbert spaces. Constr. Approx., 13(2):187-220, 1997. [PDF] Keyword(s): machine learning, neural networks, optimization, approximation theory. Abstract:

   This paper deals with sparse approximations by means of convex combinations of elements from a predetermined "basis" subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.




2. E.D. Sontag. Critical points for least-squares problems involving certain analytic functions, with applications to sigmoidal nets. Adv. Comput. Math., 5(2-3):245-268, 1996. [PDF] Keyword(s): machine learning, subanalytic sets, semianalytic sets, critical points, approximation theory, neural networks, real-analytic functions. Abstract:

   This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called "single-hidden layer neural networks", which are built upon the standard sigmoidal activation tanh(x) or equivalently 1/(1+exp(-x)), a rough upper bound for this cardinality is provided as well.

1. C. Darken, M.J. Donahue, L. Gurvits, and E.D. Sontag. Rate of approximation results motivated by robust neural network learning. In COLT '93: Proceedings of the sixth annual conference on Computational learning theory, New York, NY, USA, pages 303-309, 1993. ACM Press. [doi:http://doi.acm.org/10.1145/168304.168357] Keyword(s): machine learning, neural networks, optimization problems, approximation theory.

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