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

Emerging Convolutions for Generative Normalizing Flows by Emiel Hoogeboom, Rianne van den Verg, Max Welling Poster: Pacific Ballroom #8

Invertible functions � � dz � � p X ( x ) = p Z ( z ) � ; z = f ( x ) � � dx � • The change of variable formula holds • Admits exact log-likelihood optimization (opposed to VAEs, GANs) • Fast sampling (opposed to PixelCNNs) Emiel Hoogeboom

Background Convolutions

Background: Convolution as matrix multiplication e f h i 0 d e f g h i 1 d e g h 2 a b c 0 1 2 b c e f h i 3 a b c d e f g h i d e f 3 4 5 4 b c d e g h 5 g h i 6 7 8 b c e f 6 a b c d e f 7 b c d e 8 • Let w be a kernel, and x a feature map • A convolution is equivalent to a matrix multiplication

Convolution as matrix multiplication e f h i 0 d e f g h i 1 d e g h 2 a b c 0 1 2 b c e f h i 3 a b c d e f g h i d e f 3 4 5 4 b c d e g h 5 g h i 6 7 8 b c e f 6 a b c d e f 7 b c d e 8 in 2 in 1 out 1 out 2

Autoregressive Convolutions in 2 in 1 out 1 out 2 Standard convolution

Autoregressive Convolutions in 2 in 2 in 1 in 1 out 1 out 2 out 1 out 2 Standard convolution Autoregressive convolution • Tractable Jacobian determinant • Straightforward to invert

Method Emerging convolutions

Emerging Convolutions • Combine autoregressive convolutions • Special case: receptive field identical to standard convolutions Receptive fields of emerging convolutions

Emerging Convolutions Emerging convolution in 2 in 1 k 1 ⋆ ( k 2 ⋆ f ) = ( k 1 ∗ k 2 ) ⋆ f out 1 out 2 Equivalent filter Standard convolution

Method Periodic convolutions

Invertible Periodic Convolutions • Leverages the convolution theorem • The determinant and inverse are computed in frequency domain in 2 in 1 out 1 out 2

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