Parametric Manifold Learning Via Sparse Multidimensional Scaling

We propose a metric-learning framework for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We employ Siamese networks to solve the problem of least squares multidimensional scaling for generating mappings that preserve geodesic distances on the manifold. In contrast to previous parametric manifold learning methods we show a substantial reduction in training effort enabled by the computation of geodesic distances in a farthest point sampling strategy. Additionally, the use of a network to model the distance-preserving map reduces the complexity of the multidimensional scaling problem and leads to an improved non-local generalization of the manifold compared to analogous non-parametric counterparts. We demonstrate our claims on point-cloud data and on image manifolds and show a numerical analysis of our technique to facilitate a greater understanding of the representational power of neural networks in modeling manifold data.