Efficient, sparse representation of manifold distance matrices for classical scaling
This enables analyses of large point sets in 3-D shape analysis that were previously infeasible, addressing a domain-specific bottleneck.
The paper tackles the problem of efficiently representing geodesic distance matrices for large point sets by introducing a sparse method using biharmonic interpolation, achieving 2x faster computation and 20x less memory usage compared to leading methods while maintaining similar quality.
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.