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Non-parametric Models for Non-negative Functions

Ulysse Marteau-Ferey 1, 2, 3 Francis Bach 1, 2, 3 Alessandro Rudi 1, 2, 3
3 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple quantile regression.
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https://hal.inria.fr/hal-02891640
Contributor : Ulysse Marteau-Ferey <>
Submitted on : Tuesday, July 7, 2020 - 9:11:34 AM
Last modification on : Tuesday, May 4, 2021 - 2:06:03 PM
Long-term archiving on: : Friday, November 27, 2020 - 12:11:33 PM

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  • HAL Id : hal-02891640, version 1
  • ARXIV : 2007.03926

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Ulysse Marteau-Ferey, Francis Bach, Alessandro Rudi. Non-parametric Models for Non-negative Functions. 2020. ⟨hal-02891640⟩

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