Low-rank Characteristic Tensor Density Estimation Part I: Foundations

Effective non-parametric density estimation is a key challenge in high-dimensional multivariate data analysis. In this paper,we propose a novel approach that builds upon tensor factorization tools. Any multivariate density can be represented by its characteristic function, via the Fourier transform....

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Amiridi, Magda, Kargas, Nikos, Sidiropoulos, Nicholas D
Format: Article
Language:English
Subjects:
Characteristic functions
Data analysis
Density
Dimensional analysis
Fourier transforms
Multivariate analysis
Smoothness
Tensors
Online Access:Get full text
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Summary:Effective non-parametric density estimation is a key challenge in high-dimensional multivariate data analysis. In this paper,we propose a novel approach that builds upon tensor factorization tools. Any multivariate density can be represented by its characteristic function, via the Fourier transform. If the sought density is compactly supported, then its characteristic function can be approximated, within controllable error, by a finite tensor of leading Fourier coefficients, whose size de-pends on the smoothness of the underlying density. This tensor can be naturally estimated from observed realizations of the random vector of interest, via sample averaging. In order to circumvent the curse of dimensionality, we introduce a low-rank model of this characteristic tensor, which significantly improves the density estimate especially for high-dimensional data and/or in the sample-starved regime. By virtue of uniqueness of low-rank tensor decomposition, under certain conditions, our method enables learning the true data-generating distribution. We demonstrate the very promising performance of the proposed method using several measured datasets.
ISSN:2331-8422
DOI:10.48550/arxiv.2008.12315