DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling

TL;DR

This paper introduces DIMAL, a deep learning method that learns low-dimensional, distance-preserving embeddings of manifolds using sparse geodesic sampling, improving generalization over traditional methods.

Contribution

It presents a novel unsupervised deep learning approach combining Siamese networks with multidimensional scaling for isometric manifold embedding.

Findings

Enhanced local and nonlocal generalization compared to non-parametric methods

Effective low-dimensional embeddings from sparse landmark sampling

Provides geometric insights into deep learning generalization

Abstract

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.