# On the use of the Gram matrix for multivariate functional principal components analysis

arXiv: https://arxiv.org/abs/2306.12949 · HTML: https://mlsanity.com/paper/2306.12949/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 22 Jun 2023.

## Builds on (most cited first, 5 of 5)

- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [On Mahalanobis distance in functional settings](https://mlsanity.com/paper/1803.06550/) (2018)
- [A k-means procedure based on a Mahalanobis type distance for clustering multivariate functional data](https://mlsanity.com/paper/1708.00386/) (2017)
- [Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data](https://mlsanity.com/paper/1701.06263/) (2017)
- [New Perspectives on Centering](https://mlsanity.com/paper/2103.12176/) (2021)

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

- [DeepWeightFlow: Re-Basined Flow Matching for Generating Neural Network Weights](https://mlsanity.com/paper/2601.05052/) (2026)
- [Hierarchical Probabilistic Principal Component Analysis of Longitudinal Data](https://mlsanity.com/paper/2604.22015/) (2026)

## Abstract (arXiv)

Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability.