DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This work addresses the challenge of generating accurate manifold embeddings with limited data, offering insights into the geometric aspects of deep learning generalizability, though it appears incremental in combining existing techniques.
The paper tackles the problem of computing distance-preserving maps for low-dimensional embeddings of manifolds using an unsupervised deep learning approach, achieving significantly improved generalization in isometric mapping compared to non-parametric methods.
This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.