The Convolution Exponential and Generalized Sylvester Flows

Example code.

Below an example of the implementation of the convolution exponential. Note the very cool property that inv_conv_exp is computed by simply negating the kernel. For more detailed code see here.

import torch.nn.functional as F


def conv_exp(input, kernel, terms=10):
    B, C, H, W = input.size()

    result = input
    product = input

    for i in range(1, terms + 1):
        product = F.conv2d(product, kernel, padding=1, stride=(1, 1)) / i
        result = result + product

    return result


def inv_conv_exp(input, kernel, terms=10):
    return conv_exp(input, -kernel, terms)

Abstract

This paper introduces a new method to build linear flows, by taking the exponential of a linear transformation. This linear transformation does not need to be invertible itself, and the exponential has the following desirable properties: it is guaranteed to be invertible, its inverse is straightforward to compute and the log Jacobian determinant is equal to the trace of the linear transformation. An important insight is that the exponential can be computed implicitly, which allows the use of convolutional layers. Using this insight, we develop new invertible transformations named convolution exponentials and graph convolution exponentials, which retain the equivariance of their underlying transformations. In addition, we generalize Sylvester Flows and propose Convolutional Sylvester Flows which are based on the generalization and the convolution exponential as basis change. Empirically, we show that the convolution exponential outperforms other linear transformations in generative flows on CIFAR10 and the graph convolution exponential improves the performance of graph normalizing flows. In addition, we show that Convolutional Sylvester Flows improve performance over residual flows as a generative flow model measured in log-likelihood.

Experiments

The commands below reproduce the results for the mixing experiments:

python main_experiment.py --n_levels 2 --densenet_depth 8 --n_subflows 10 --densenet_growth 64 --n_intermediate_channels 64 --use_splitprior --dequantize_distribution flow --use_gated_conv --epochs 1000 --exp_name 'conv1x1' --mixing '1x1'

python main_experiment.py --n_levels 2 --densenet_depth 8 --n_subflows 10 --densenet_growth 63 --n_intermediate_channels 64 --use_splitprior --dequantize_distribution flow --use_gated_conv --epochs 1000 --exp_name 'convexp' --mixing 'convexp'

python main_experiment.py --n_levels 2 --densenet_depth 8 --n_subflows 10 --densenet_growth 63 --n_intermediate_channels 64 --use_splitprior --dequantize_distribution flow --use_gated_conv --epochs 1000 --exp_name 'emerging' --mixing 'emerging'

python main_experiment.py --n_levels 2 --densenet_depth 8 --n_subflows 10 --densenet_growth 63 --n_intermediate_channels 64 --use_splitprior --dequantize_distribution flow --use_gated_conv --epochs 1000 --exp_name 'woodbury' --mixing 'woodbury'

The Commands below reproduce the results for the residual transformations experiment:

Conv. Sylvester Flow (var. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_internal_channels 528 --use_splitprior --dequantize_distribution flow --epochs 1000 --exp_name 'sylvester' --model_type sylvester

Conv. Sylvester Flow (unif. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_internal_channels 528 --use_splitprior --dequantize_distribution uniform --epochs 1000 --exp_name 'sylvester' --model_type sylvester

Residual Block Flow (var. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution flow --epochs 1000 --exp_name 'residualflow' --model_type residual_flow

Residual Block Flow (unif. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution uniform --epochs 1000 --exp_name 'residualflow' --model_type residual_flow

Baseline Coupling Flow (var. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution flow --epochs 1000 --exp_name 'coupling' --model_type coupling_flow

Baseline Coupling Flow (unif. deq.):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 16 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution uniform --epochs 1000 --exp_name 'coupling' --model_type coupling_flow

Residual Block Flow (var. deq.) (equal memory):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 8 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution flow --epochs 1000 --exp_name 'residualflow' --model_type residual_flow

Residual Block Flow (unif. deq.) (equal memory):

python experiment_sylvester_network.py --n_levels 3 --n_subflows 8 --n_intermediate_channels 528 --use_splitprior --dequantize_distribution uniform --epochs 1000 --exp_name 'residualflow' --model_type residual_flow

Acknowledgements

The Robert Bosch GmbH is acknowledged for financial support.