# A dual basis approach to multidimensional scaling

arXiv: https://arxiv.org/abs/2303.05682 · HTML: https://mlsanity.com/paper/2303.05682/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 20 references that are in the map.
- First posted on arXiv 10 Mar 2023.

## Builds on (most cited first, 1 of 1)

- [Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion](https://mlsanity.com/paper/1804.04310/) (2018)

## Built on by (most cited first, 5 of 5)

- [Localization from structured distance matrices via low-rank matrix recovery](https://mlsanity.com/paper/2311.18076/) (2023)
- [Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees](https://mlsanity.com/paper/2410.06376/) (2024)
- [Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization](https://mlsanity.com/paper/2410.16982/) (2024)
- [Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent](https://mlsanity.com/paper/2504.19530/) (2025)
- [Provable Non-Convex Euclidean Distance Matrix Completion: Geometry, Reconstruction, and Robustness](https://mlsanity.com/paper/2508.00091/) (2025)

## Abstract (arXiv)

Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.