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arXiv · 1711.06011

DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling

Gautam Pai·Ronen Talmon·Alex Bronstein·Ron Kimmel
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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.

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Gautam Pai, Ronen Talmon, Alex Bronstein, Ron Kimmel. 2017-11-16. DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling. https://arxiv.org/abs/1711.06011

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