# Efficient, sparse representation of manifold distance matrices for classical scaling

## TL;DR

This paper introduces a novel sparse method using biharmonic interpolation to efficiently approximate geodesic distance matrices, enabling faster analysis of large 3D shape datasets with reduced memory usage.

## Contribution

A new sparse representation technique for geodesic distance matrices that significantly improves speed and memory efficiency in multidimensional scaling.

## Key findings

- 2x faster than existing methods
- 20x less memory usage
- Maintains similar approximation quality

## Abstract

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using methods such as multidimensional scaling (MDS). In this paper we present a novel sparse method for efficiently representing geodesic distance matrices using biharmonic interpolation. This method exploits knowledge of the data manifold to learn a sparse interpolation operator that approximates distances using a subset of points. We show that our method is 2x faster and uses 20x less memory than current leading methods for solving MDS on large point sets, with similar quality. This enables analyses of large point sets that were previously infeasible.