DeepAI AI Chat
Log In Sign Up

Matrix factorisation and the interpretation of geodesic distance

06/02/2021
by   Nick Whiteley, et al.
0

Given a graph or similarity matrix, we consider the problem of recovering a notion of true distance between the nodes, and so their true positions. Through new insights into the manifold geometry underlying a generic latent position model, we show that this can be accomplished in two steps: matrix factorisation, followed by nonlinear dimension reduction. This combination is effective because the point cloud obtained in the first step lives close to a manifold in which latent distance is encoded as geodesic distance. Hence, a nonlinear dimension reduction tool, approximating geodesic distance, can recover the latent positions, up to a simple transformation. We give a detailed account of the case where spectral embedding is used, followed by Isomap, and provide encouraging experimental evidence for other combinations of techniques.

READ FULL TEXT

page 1

page 2

page 3

page 4

06/09/2020

Manifold structure in graph embeddings

Statistical analysis of a graph often starts with embedding, the process...
07/29/2011

A Invertible Dimension Reduction of Curves on a Manifold

In this paper, we propose a novel lower dimensional representation of a ...
06/16/2008

Local Procrustes for Manifold Embedding: A Measure of Embedding Quality and Embedding Algorithms

We present the Procrustes measure, a novel measure based on Procrustes r...
12/22/2020

Unsupervised Functional Data Analysis via Nonlinear Dimension Reduction

In recent years, manifold methods have moved into focus as tools for dim...
07/13/2016

Fitting a Simplicial Complex using a Variation of k-means

We give a simple and effective two stage algorithm for approximating a p...
05/21/2013

Out-of-sample Extension for Latent Position Graphs

We consider the problem of vertex classification for graphs constructed ...