# CovNet: Covariance Networks for Functional Data on Multidimensional Domains

arXiv: https://arxiv.org/abs/2104.05021 · HTML: https://mlsanity.com/paper/2104.05021/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 51 references that are in the map.
- First posted on arXiv 11 Apr 2021.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Smooth function approximation by deep neural networks with general activation functions](https://mlsanity.com/paper/1906.06903/) (2019)
- [Approximating smooth functions by deep neural networks with sigmoid activation function](https://mlsanity.com/paper/2010.04596/) (2020)
- [Low-Rank Covariance Function Estimation for Multidimensional Functional Data](https://mlsanity.com/paper/2008.12919/) (2020)
- [Estimation of the Mean Function of Functional Data via Deep Neural Networks](https://mlsanity.com/paper/2012.04573/) (2020)
- [Tests for separability in nonparametric covariance operators of random surfaces](https://mlsanity.com/paper/1505.02023/) (2015)
- [Testing separability of space--time functional processes](https://mlsanity.com/paper/1509.07017/) (2015)
- [A Test for Separability in Covariance Operators of Random Surfaces](https://mlsanity.com/paper/1710.08388/) (2017)
- [Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula](https://mlsanity.com/paper/1901.06386/) (2019)
- [Separable Expansions for Covariance Estimation](https://mlsanity.com/paper/2007.12175/) (2020)

## Built on by (most cited first, 4 of 4)

- [Online Learning of Functional Principal Component Analysis for Multidimensional Functional Data](https://mlsanity.com/paper/2505.02131/) (2025)
- [Deep learning estimation of the spectral density of functional time series on large domains](https://mlsanity.com/paper/2601.00284/) (2026)
- [A Theory of Nonparametric Covariance Function Estimation for Discretely Observed Data](https://mlsanity.com/paper/2603.23302/) (2026)
- [Deep Shape Regression for Planar Curves with Multimodal Covariates](https://mlsanity.com/paper/2607.19600/) (2026)

## Abstract (arXiv)

Covariance estimation is ubiquitous in functional data analysis. Yet, the case of functional observations over multidimensional domains introduces computational and statistical challenges, rendering the standard methods effectively inapplicable. To address this problem, we introduce "Covariance Networks" (CovNet) as a modeling and estimation tool. The CovNet model is "universal" - it can be used to approximate any covariance up to desired precision. Moreover, the model can be fitted efficiently to the data and its neural network architecture allows us to employ modern computational tools in the implementation. The CovNet model also admits a closed-form eigendecomposition, which can be computed efficiently, without constructing the covariance itself. This facilitates easy storage and subsequent manipulation of a covariance in the context of the CovNet. We establish consistency of the proposed estimator and derive its rate of convergence. The usefulness of the proposed method is demonstrated by means of an extensive simulation study and an application to resting state fMRI data.