# Functional Regression

arXiv: https://arxiv.org/abs/1406.4068 · HTML: https://mlsanity.com/paper/1406.4068/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2014.

## Built on by (most cited first, 26 of 26)

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- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Non-linear Functional Modeling using Neural Networks](https://mlsanity.com/paper/2104.09371/) (2021)
- [Modern Non-Linear Function-on-Function Regression](https://mlsanity.com/paper/2107.14151/) (2021)
- [Statistical Learning and Inverse Problems: A Stochastic Gradient Approach](https://mlsanity.com/paper/2209.14967/) (2022)
- [Neural Networks for Scalar Input and Functional Output](https://mlsanity.com/paper/2208.05776/) (2022)
- [Spherical Analysis of Learning Nonlinear Functionals](https://mlsanity.com/paper/2410.01047/) (2024)
- [Learning the Dynamics of Sparsely Observed Interacting Systems](https://mlsanity.com/paper/2301.11647/) (2023)
- [Functional additive models on manifolds of planar shapes and forms](https://mlsanity.com/paper/2109.02624/) (2021)
- [Error-in-variables modelling for operator learning](https://mlsanity.com/paper/2204.10909/) (2022)
- [Interpretable Causal Inference for Analyzing Wearable, Sensor, and Distributional Data](https://mlsanity.com/paper/2312.10569/) (2023)
- [Functional Output Regression with Infimal Convolution: Exploring the Huber and ε-insensitive Losses](https://mlsanity.com/paper/2206.08220/) (2022)
- [Maximally Forward-Looking Core Inflation](https://mlsanity.com/paper/2404.05209/) (2024)
- [Statistical Opportunities in Neuroimaging](https://mlsanity.com/paper/2602.12974/) (2026)
- [Bayesian adaptive and interpretable functional regression for exposure profiles](https://mlsanity.com/paper/2203.00784/) (2022)
- [Classification ensembles for multivariate functional data with application to mouse movements in web surveys](https://mlsanity.com/paper/2205.13380/) (2022)
- [Industrial Data Science for Batch Manufacturing Processes](https://mlsanity.com/paper/2209.09660/) (2022)
- [Enhancing Functional Data Analysis with Sequential Neural Networks: Advantages and Comparative Study](https://mlsanity.com/paper/2311.01875/) (2023)
- [Heterogeneous Image-based Classification Using Distributional Data Analysis](https://mlsanity.com/paper/2403.07126/) (2024)
- [Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression](https://mlsanity.com/paper/2502.16888/) (2025)
- [MAGIC: Multi-task Gaussian process for joint imputation and classification in healthcare time series](https://mlsanity.com/paper/2509.19577/) (2025)
- [Deep Gaussian Processes for Functional Maps](https://mlsanity.com/paper/2510.22068/) (2025)
- [Bayesian Optimization for Function-Valued Responses under Min-Max Criteria](https://mlsanity.com/paper/2512.07868/) (2025)
- [Dynamic Frechet Regression with Feature Selection for Distributional Data](https://mlsanity.com/paper/2607.09613/) (2026)
- [Doubly Robust Functional Representation Learning for Longitudinal Causal Inference with Irregular Histories](https://mlsanity.com/paper/2607.28567/) (2026)
- [Functional Attentive Interpretable Regression](https://mlsanity.com/paper/2609.05846/) (2026)

## Abstract (arXiv)

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field.