# Low-rank Characteristic Tensor Density Estimation Part I: Foundations

arXiv: https://arxiv.org/abs/2008.12315 · HTML: https://mlsanity.com/paper/2008.12315/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2020.

## Built on by (most cited first, 15 of 15)

- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Beyond Smoothness: Incorporating Low-Rank Analysis into Nonparametric Density Estimation](https://mlsanity.com/paper/2204.00930/) (2022)
- [Sample Complexity Using Infinite Multiview Models](https://mlsanity.com/paper/2302.04292/) (2023)
- [Scale adaptive and robust intrinsic dimension estimation via optimal neighbourhood identification](https://mlsanity.com/paper/2405.15132/) (2024)
- [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024)
- [Generalized Identifiability Bounds for Mixture Models with Grouped Samples](https://mlsanity.com/paper/2207.11164/) (2022)
- [Generalization Bounds for Inductive Matrix Completion in Low-noise Settings](https://mlsanity.com/paper/2212.08339/) (2022)
- [TERM Model: Tensor Ring Mixture Model for Density Estimation](https://mlsanity.com/paper/2312.08075/) (2023)
- [Joint Probability Estimation Using Tensor Decomposition and Dictionaries](https://mlsanity.com/paper/2203.01667/) (2022)
- [Estimating Joint Probability Distribution With Low-Rank Tensor Decomposition, Radon Transforms and Dictionaries](https://mlsanity.com/paper/2304.08740/) (2023)
- [PMODE: Theoretically Grounded and Modular Mixture Modeling](https://mlsanity.com/paper/2508.21396/) (2025)
- [Learning Multivariate CDFs and Copulas using Tensor Factorization](https://mlsanity.com/paper/2210.07132/) (2022)
- [A Unified MDL-based Binning and Tensor Factorization Framework for PDF Estimation](https://mlsanity.com/paper/2504.18686/) (2025)
- [Generalization Bounds for Semi-supervised Matrix Completion with Distributional Side Information](https://mlsanity.com/paper/2511.13049/) (2025)

## Abstract (arXiv)

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.