# Regression with Distance Matrices

arXiv: https://arxiv.org/abs/1303.3750 · HTML: https://mlsanity.com/paper/1303.3750/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2013.

## Built on by (most cited first, 8 of 8)

- [Deep Fr'echet Regression](https://mlsanity.com/paper/2407.21407/) (2024)
- [End-to-End Deep Learning for Predicting Metric Space-Valued Outputs](https://mlsanity.com/paper/2509.23544/) (2025)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Fr'echet Geodesic Boosting](https://mlsanity.com/paper/2509.18013/) (2025)
- [FDRMFL: Multimodal Federated Feature Extraction Model Based on Information Maximization and Contrastive Learning](https://mlsanity.com/paper/2512.02076/) (2025)
- [Deep Single-Index Fr'echet Regression](https://mlsanity.com/paper/2606.06957/) (2026)
- [Dynamic Frechet Regression with Feature Selection for Distributional Data](https://mlsanity.com/paper/2607.09613/) (2026)

## Abstract (arXiv)

Data types that lie in metric spaces but not in vector spaces are difficult to use within the usual regression setting, either as the response and/or a predictor. We represent the information in these variables using distance matrices which requires only the specification of a distance function. A low-dimensional representation of such distance matrices can be obtained using methods such as multidimensional scaling. Once these variables have been represented as scores, an internal model linking the predictors and the response can be developed using standard methods. We call scoring the transformation from a new observation to a score while backscoring is a method to represent a score as an observation in the data space. Both methods are essential for prediction and explanation. We illustrate the methodology for shape data, unregistered curve data and correlation matrices using motion capture data from an experiment to study the motion of children with cleft lip.