On the Representation of Continuous Functions of Several Variables as Superpositions of Continuous Functions of one Variable and Addition

By V. Tikhomirov. 1991

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Citations in the corpus

Deep learning in neural networks: An overview (2014, background): “…of a solution candidate by the length of the shortest program that computes it (e.g., Blumer, Ehrenfeucht, Haussler, & Warmuth, 1987; Chaitin, 1966; Grünwald,Myung, & Pitt, 2005; Kolmogorov, 1965b; Levin, 1973a; Li & Vitányi, 1997; Rissanen, 1986; Solomonoff, 1964, 1978;Wallace & Boulton, 1968).”
On the approximate realization of continuous mappings by neural networks (1989, methodology): “Therefore we show another proof for the four-layer case by using the Kolmogorov-Arnold-Sprecher theorem (Kolmogorov, 1957; Sprecher, 1965).”
KAN: Kolmogorov-Arnold Networks (2024, unknown): “Whereas MLPs are inspired by the universal approximation theorem, KANs are inspired by the Kolmogorov-Arnold representation theorem [6, 7].”
Theory of classification : a survey of some recent advances (2005, methodology): “The chaining idea appears in the work of Kolmogorov [122, 123] and was further developed by Dudley [81] and Pollard [180].”
Bond Risk Premia with Machine Learning∗ (2018, unknown): “The result is a powerful learning method which can approximate virtually any continuous function with compact support (see Kolmogorov, 1957; Diaconis and Shahshahani, 1984; Cybenko, 1989; Hornik, Stinchcombe and White, 1989).”
Artificial neural networks for computer-based molecular design. (1998, background): “They follow the principle of convoluting simple nonlinear functions for approximation of complicated input±output relationships (Kolmogorov's theorem, Kolmogorov (1957)).”
On a Constructive Proof of Kolmogorov’s Superposition Theorem (2009, methodology): “In 1957 the Russian mathematician Kolmogorov showed the remarkable fact that any continuous function f of many variables can be represented as a composition of addition and some functions of one variable [14].”
Smooth function approximation using neural networks (2005, background): “Another line of research has focused on the mathematical investigation of networks' approximation properties [19]–[22].”
Posterior Concentration for Sparse Deep Learning (2018, methodology): “Kolmogorov (1963) provided another motivation for deep networks by showing that superpositions of univariate semi-affine functions provide a universal basis for representing multivariate functions.”
Quantum Entanglement in Neural Network States (2017, background): “…community RBMs have had successful applications in classiﬁcation [50], dimensionality reduction [48], feature learning [56], and collaborative ﬁltering [49], etc. Math-ematically, the ability of the RBM to approximate any many-body state is assured by representability theorems [57–59].”
Online Prediction of Time Series Data With Kernels (2009, background): “They are proven to be universal approximators under suitable conditions, thus providing the means to capture information in data that is difficult to identify using other techniques [9].”
Handbook of Granular Computing (2008, methodology): “The meaning of arithmetic depends directly on the extension principle in force since arithmetic operations are (continuous) functions over the reals assuming that division by zero is not allowed and over the extended reals [31] when division by zero is allowed.”