Emerging Convolutions for Generative Normalizing Flows by Emiel - - PowerPoint PPT Presentation

Emerging Convolutions

by Emiel Hoogeboom, Rianne van den Verg, Max Welling

for Generative Normalizing Flows

Poster: Pacific Ballroom #8

Invertible functions

Emiel Hoogeboom

pX(x) = pZ(z)

• dz

dx

• ; z = f(x)
• The change of variable formula holds
• Admits exact log-likelihood optimization (opposed to VAEs, GANs)
• Fast sampling (opposed to PixelCNNs)

Background

Convolutions

Background: Convolution as matrix multiplication

a b c d e f g h i 1 2 3 4 5 6 7 8

h i h g h g i h i h g h g i d e e e f f d d e e e f f d d e e e f f d b c b a c b c b c b a c b c 1 2 3 4 5 6 7 8

• Let w be a kernel, and x a feature map
• A convolution is equivalent to a matrix multiplication

Convolution as matrix multiplication

a b c d e f g h i 1 2 3 4 5 6 7 8

h i h g h g i h i h g h g i d e e e f f d d e e e f f d d e e e f f d b c b a c b c b c b a c b c 1 2 3 4 5 6 7 8

• ut 1

in 1 in 2

• ut 2

Autoregressive Convolutions

• ut 1

in 1 in 2

• ut 2

Standard convolution

Autoregressive Convolutions

• ut 1

in 1 in 2

• ut 2

Standard convolution

• Tractable Jacobian determinant
• Straightforward to invert
• ut 1

in 1 in 2

• ut 2

Autoregressive convolution

Method

Emerging convolutions

Emerging Convolutions

Receptive fields of emerging convolutions

• Combine autoregressive convolutions
• Special case: receptive field identical to

standard convolutions

Emerging Convolutions

• ut 1

in 1 in 2

• ut 2

Standard convolution Emerging convolution

k1 ⋆ (k2 ⋆ f) = (k1 ∗ k2) ⋆ f

Equivalent filter

Method

Periodic convolutions

Invertible Periodic Convolutions

• ut 1

in 1 in 2

• ut 2
• Leverages the convolution theorem
• The determinant and inverse are computed in frequency domain

Conclusion

• Emerging convolutions
• Invertible periodic convolutions
• Stable, flexible 1x1 QR convolutions
• Poster #8