# An Information Geometry of Statistical Manifold Learning

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Ke Sun,
Stéphane Marchand-Maillet
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):1-9, 2014.

#### Abstract

Manifold learning seeks low-dimensional representations of high-dimensional data. The main tactics have been exploring the geometry in an input data space and an output embedding space. We develop a manifold learning theory in a hypothesis space consisting of models. A model means a specific instance of a collection of points, e.g., the input data collectively or the output embedding collectively. The semi-Riemannian metric of this hypothesis space is uniquely derived in closed form based on the information geometry of probability distributions. There, manifold learning is interpreted as a trajectory of intermediate models. The volume of a continuous region reveals an amount of information. It can be measured to define model complexity and embedding quality. This provides deep unified perspectives of manifold learning theory.

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