--- date: '2025-11-23' description: a form of non-linear dimensionality reduction id: MDS modified: 2026-06-05 15:08:05 GMT-04:00 tags: - ml title: multidimensional scaling created: '2025-11-23' published: '2025-11-23' pageLayout: default slug: thoughts/MDS permalink: https://aarnphm.xyz/thoughts/MDS.md generator: quartz: v4.6.0 hostedProvider: Cloudflare baseUrl: aarnphm.xyz full: https://aarnphm.xyz/llms-full.txt --- _refers to a set of related ordination techniques in information visualisation_ Takes a some high-dimensional data $x_{1},\dots,x_{n} \in \mathbb{R}^{d}$ and produce a lower-dimensional representation $y_{1},\dots,y_{n} \in \mathbb{R}^{d}$ where pairwise distances (represented by a matrix $D \in \mathbb{R}^{d\times d}$ with $D_{ij}$ corresponds to some measure of distance between points $x_{i}$ and $x_{j}$) in classical MDS, we use Euclidean distance metrics $d_{ij} = \lVert x_{i} - x_{j} \rVert_{2}$ - MDS then seek to find points $y_{1},\dots,y_{n}\in \mathbb{R}^{p}$ such that the distances between these new points approximate to the original distance $\lVert y_{i} - y_{j} \rVert{2} \approx d_{ij}$ the sketch below shows a toy Euclidean distance graph (left) and its 2D embedding produced by classical MDS (right); edge labels are the target distances $d_{ij}$ the embedding tries to preserve.