Geometric View of GAN and Visualization

Zhizhong Li

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May 5, 2017, CUHK

Generative Adversarial Networks (GAN)

• A powerful generative model

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• z: noise from some distribution�x hat: generated fake sample�x: real sample

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Manifold - Fundamental Concept in Geometry

• Locally resembles Euclidean space
• Parametrized by coordinate systems
• Surface in R^n

GAN as Manifold Learning

• Learn a coordinate system that represents data manifold

Generator

Latent space

Data manifold

Related Work

• The idea appears implicitly in many works
• WGAN use geometric interpretation to explain difficulty in training
• Mode regularized GAN introduces manifold-diffusion training
• T. White uses spheres as latent space

Impact of Geometric Properties

• Dimensionality�What would happen to GAN if the dimensions of latent space and data manifold do not match?
• Connectivity�If the data manifold may have several disconnected components. Do we have to make the latent space disconnected at the same time?

Dimensionality

1 dim spiral -> 2 dim gaussian

2 dim gaussian -> 1 dim spiral

Dimensionality

• Dim z < Dim x�the generated low dimensional manifold try to cover the larger dimensional data manifold as possible.
• Dim z > Dim x�then the generated manifold is able to cover the data manifold.�However, it may generate many extra fake examples.

Connectivity

1 square -> 9 squares

9 squares -> 1 square

Connectivity

• Connected z -> Disconnected x�easy to learn.
• Disconnected x -> Connected z�Cannot cover the whole data manifold.

Thanks!

Codes

https://github.com/innerlee/ELEG5491/

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Zhizhong Li

lz015@ie.cuhk.edu.hk

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The Chinese University of Hong Kong

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Future Study

• Decouple Geometry and Distribution
• Inverse Mapping
• Learning Mapping as Graph