# Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems

arXiv: https://arxiv.org/abs/2107.14728 · HTML: https://mlsanity.com/paper/2107.14728/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 78 references that are in the map.
- First posted on arXiv 30 Jul 2021.

## Builds on (most cited first, 10 of 10)

- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization](https://mlsanity.com/paper/1506.04209/) (2015)
- [A least-squares method for sparse low rank approximation of multivariate functions](https://mlsanity.com/paper/1305.0030/) (2013)
- [Low-Rank Covariance Function Estimation for Multidimensional Functional Data](https://mlsanity.com/paper/2008.12919/) (2020)
- [Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes](https://mlsanity.com/paper/1910.03134/) (2019)
- [Sparse and Functional Principal Components Analysis](https://mlsanity.com/paper/1309.2895/) (2013)
- [Multi-dimensional Functional Principal Component Analysis](https://mlsanity.com/paper/1510.04439/) (2015)
- [Tests for separability in nonparametric covariance operators of random surfaces](https://mlsanity.com/paper/1505.02023/) (2015)
- [A test of weak separability for multi-way functional data, with application to brain connectivity studies](https://mlsanity.com/paper/1703.10210/) (2017)
- [Fast Covariance Estimation for Multivariate Sparse Functional Data](https://mlsanity.com/paper/1812.00538/) (2018)

## Built on by (most cited first, 2 of 2)

- [scikit-fda: A Python Package for Functional Data Analysis](https://mlsanity.com/paper/2211.02566/) (2022)
- [NeuroPMD: Neural Fields for Density Estimation on Product Manifolds](https://mlsanity.com/paper/2501.02994/) (2025)

## Abstract (arXiv)

Many modern datasets, from areas such as neuroimaging and geostatistics, come in the form of a random sample of tensor-valued data which can be understood as noisy observations of a smooth multidimensional random function. Most of the traditional techniques from functional data analysis are plagued by the curse of dimensionality and quickly become intractable as the dimension of the domain increases. In this paper, we propose a framework for learning continuous representations from a sample of multidimensional functional data that is immune to several manifestations of the curse. These representations are constructed using a set of separable basis functions that are defined to be optimally adapted to the data. We show that the resulting estimation problem can be solved efficiently by the tensor decomposition of a carefully defined reduction transformation of the observed data. Roughness-based regularization is incorporated using a class of differential operator-based penalties. Relevant theoretical properties are also established. The advantages of our method over competing methods are demonstrated in a simulation study. We conclude with a real data application in neuroimaging.