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Asymptotic Convergence of Soft-Constrained Neural Networks for Density Estimation

Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy
Mathematics 2020, 8(4), 572; https://doi.org/10.3390/math8040572
Received: 7 March 2020 / Revised: 7 April 2020 / Accepted: 8 April 2020 / Published: 12 April 2020
(This article belongs to the Special Issue Neural Networks and Learning Systems)
A soft-constrained neural network for density estimation (SC-NN-4pdf) has recently been introduced to tackle the issues arising from the application of neural networks to density estimation problems (in particular, the satisfaction of the second Kolmogorov axiom). Although the SC-NN-4pdf has been shown to outperform parametric and non-parametric approaches (from both the machine learning and the statistics areas) over a variety of univariate and multivariate density estimation tasks, no clear rationale behind its performance has been put forward so far. Neither has there been any analysis of the fundamental theoretical properties of the SC-NN-4pdf. This paper narrows the gaps, delivering a formal statement of the class of density functions that can be modeled to any degree of precision by SC-NN-4pdfs, as well as a proof of asymptotic convergence in probability of the SC-NN-4pdf training algorithm under mild conditions for a popular class of neural architectures. These properties of the SC-NN-4pdf lay the groundwork for understanding the strong estimation capabilities that SC-NN-4pdfs have only exhibited empirically so far. View Full-Text
Keywords: soft-constrained neural network; probabilistic interpretation of neural networks; density estimation; nonpaltry density function soft-constrained neural network; probabilistic interpretation of neural networks; density estimation; nonpaltry density function
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Trentin, E. Asymptotic Convergence of Soft-Constrained Neural Networks for Density Estimation. Mathematics 2020, 8, 572.

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