This is the Framework for Easily Invertible Architectures (FrEIA).

• Construct Invertible Neural Networks (INNs) from simple invertible building blocks.
• Quickly construct complex invertible computation graphs and INN topologies.
• Forward and inverse computation guaranteed to work automatically.
• Most common invertible transforms and operations are provided.
• Easily add your own invertible transforms.

Table of contents

• Papers
• Installation
  • Dependencies
  • Downloading + Installing
  • Documentation
• Tutorial
  • Basic Concepts
  • Invertible Computation Graphs
  • Node Construction
    • General API
    • Affine Coupling Blocks
  • Examples
    • Simple GLOW-based INN
    • Conditional INN
    • Convolutional INN
  • Useful Tips & Engineering Heuristics

Papers

Our following papers use FrEIA, with links to code given below.

"Exact Information Bottleneck with Invertible Neural Networks: Getting the Best of Discriminative and Generative Modeling" (2020)

• Paper: https://arxiv.org/abs/2001.06448
• Code: https://github.com/VLL-HD/exact_information_bottleneck

"Guided Image Generation with Conditional Invertible Neural Networks" (2019)

• Paper: https://arxiv.org/abs/1907.02392
• Supplement: https://drive.google.com/file/d/1_OoiIGhLeVJGaZFeBt0OWOq8ZCtiI7li
• Code: https://github.com/VLL-HD/conditional_invertible_neural_networks

"Analyzing inverse problems with invertible neural networks." (2018)

• Paper: https://arxiv.org/abs/1808.04730
• Code: https://github.com/VLL-HD/analyzing_inverse_problems

Installation

Dependencies

Package	Version
Pythton	>= 3.7
Pytorch	>= 1.0.0
Numpy	>= 1.15.0

Downloading + Installing

To just install the framework:

pip install git+https://github.com/VLL-HD/FrEIA.git

For development:

# first clone the repository
git clone 'github.com/VLL-HD/FrEIA.git'
# then install in development mode, so that changes don't require a reinstall
cd FrEIA
python setup.py develop

Documentation

Besides the general tutorial below, the complete documentation is found in the ./docs/ directory, or at

https://vll-hd.github.io/FrEIA

Tutorial

Basic Concepts

"Why does FrEIA even exist? RealNVP can be implemented in ~100 lines of code!"

That is correct, but the concept of INNs is more general: For any computation graph, as long as each node in the graph is invertible, and there are no loose ends, the entire computation is invertible. This is also true if the operation nodes have multiple in- or outputs, e.g. concatenation (n inputs, 1 output). So we need a framework that allows to define an arbitrary computation graph, consisiting of invertible operations.

For example, consider wanting to implement some complicated new INN architecture, with multiple in- and outputs, skip connections, a conditional part, ...:

To allow efficient prototyping and experimentation with such architectures, we need a framework that can perform the following tasks:

• As the inputs of operations depend on the outputs of others, we have to infer the order of operations, both for the forward and the inverse direction.
• The operators have to be initialized with the correct input- and output sizes in mind (e.g. required number of weights), i.e. we have to perform shape inference on the computation graph.
• During the computation, we have to keep track of intermediate results (edges in the graph) and store them until they are needed.
• We want to use pytorch methods and tools, such as .cuda(), .state_dict(), DataParallel(), etc. on the entire computation graph, without worrying whether they work correctly or having to fix them.

Along with an interface to define INN computation graphs and invertible operators within, these are the main tasks that FrEIA addresses.

Invertible Computation Graphs

The building blocks of the INN computation graph are the nodes in it. They are provided through the FrEIA.framework.Node class. The computation graph is constructed by constructing each node, given its inputs (defining one direction of the INN as the 'forward' computation). More specifically:

• The Node-subclass InputNode represents an input to the INN, and its constructor only takes the dimensions of the data (except the batch dimension). E.g. for a 32x32 RGB image:

  in1 = InputNode(3, 32, 32, name='Input x_1')

  The name argument can be omitted in principle, but it is recommended in general, as it appears e.g. in error messages.

• Each Node (and derived classes) has properties node.out0, node.out1, etc., depending on its number of outputs. Instead of node.out{i}, it is equivalent to use a tuple (node, i), which is useful if you e.g. want to loop over 10 outputs of a node.

• Each Node is initialized given a list of its inputs as the first constructor argument, along with other arguments covered later (omitted as '...' in the following, in particular defining what operation the node should represent). For transform T1 in the example above, this would look like the this:

  transf1 = Node([in1.out0], ..., name='Transform T_1')

  Or for merge m2:

  merge2 = Node([transf3.out0, split2.out1], ..., name='Merge m_2')

  Conditions are passed as a list through the conditions argument:

  transf3 = Node([merge1.out0], ..., conditions=[cond], name='Transform T_3')

• The Node-subclass OutputNode is used for the outputs. The INN as a whole will return the result at this node.

• Conditions (as in the cINN paper) are represented by ConditionNode, whose constructor is identical to the InputNode.

• Take note of several features for convenience (also see examples below): 1.) If a preceding node only has a single output, it is also equivalent to directly use node instead of node.out0 in the constructor of following nodes. 2.) If a node only takes a sinlge input/condition, you can directly use only that input in the constructor instead of a list, i.e. node.out0 instead of [node.out0].

• From the list of nodes, the INN is represented by the class FrEIA.framework.ReversibleGraphNet. The constructor takes a list of all the nodes in the INN (order irrelevant), and an optional verbose argument (True by default. If verbose, the results of the shape inference as well as the in/outputs of each node are printed to stdout.)

• The ReversibleGraphNet is a subclass of torch.nn.Module, and can be used like any other torch Module. For the computation, the inputs are given as a list of torch tensors, or just a single torch tensor if there is only one input. To perform the inverse pass, the rev argument has to be set to True (see examples).

Using these rules, we would construct the INN from the above example in the following way:

in1 = InputNode(3, 32, 32, name='Input x_1') # 3-channel image
in2 = InputNode(128, name='Input x_2') # 1D vector
cond = ConditionNode(42, name='Condition c')

transf1 = Node([in1.out0], ..., name='Transform T_1')
split1 =  Node([transf1.out0], ..., name='Split s_1')
split2 =  Node([split1.out0], ..., name='Split s_2')
transf2 = Node([split2.out0], ..., name='Transform T_2')
merge1 =  Node([transf2.out0, in2.out0], ..., name='Merge m_1')
transf3 = Node([merge1.out0], ..., conditions=[cond], name='Transform T_3')
merge2 =  Node([transf2.out0, split2.out1], ..., name='Merge m_2')

output1 = Node([split1.out1], ..., name='Output z_1')
output2 = Node([merge2.out0], ..., name='Output z_2')

example_INN = ReversibleGraphNet([in1, in2, output1, output2, cond,
                                  transf1, transf2, transf3,
                                  merge1, merge2, split1, split2], verbose=False)

# dummy inputs:
x1, x2, cond = torch.randn(1,3,32,32), torch.randn(1, 128), torch.randn(1, 42)

# compute the outputs
z1, z2 = example_INN([x1, x2], c=cond)

# invert the network and check if we get the original inputs back:
x1_inv, x2_infv = example_INN([z1, z2], c=cond, rev=True)
assert (torch.max(torch.abs(x1_inv - x1)) < 1e-5
        and torch.max(torch.abs(x2_inv - x2)) < 1e-5)

Node Construction

Above, we only covered the construction of the computation graph itself, but so far we have not shown how to define the operations represented by each node. Therefore, we will take a closer look at the Node constructor and its arguments:

Node(inputs, module_type, module_args, conditions=[], name=None)

General API

The arguments of the Node constructor are the following:

• inputs: A list of outputs of other nodes, that are used as inputs for this node (discussed above)
• module_type: This argument gives the class of operation to be performed by this node, for example GLOWCouplingBlock for a coupling block following the GLOW-design. Many implemented classes can be found in the documentation under https://vll-hd.github.io/FrEIA/modules/index.html
• module_args: This argument is a dictionary. It provides arguments for the module_type-constructor. For instance, a random invertible permutation (module_type=PermuteLayer) only has one argument seed, so we could use module_args={'seed':111}.

Affine Coupling Blocks

All coupling blocks (GLOW, RNVP, NICE), merit special discussion, because they are the most used invertible transforms.

• The coupling blocks contain smaller feed-forward subnetworks predicting the affine coefficients. The in- and output shapes of the subnetworks depend on the in- output size of the coupling block itself. These size are not known when coding the INN (or perhaps can be worked out by hand, but would have to be worked out anew every time the architecture is modified slightly). Therefore, the subnetworks can not be directly passed as nn.Modules, but rather in the form of a function or class, that constructs the subnetworks given in- and output size. This is a lot simpler than it sounds, for a fully connected subnetwork we could use e.g.

  def fc_constr(c_in, c_out):
      return nn.Sequential(nn.Linear(c_in, 128), nn.ReLU(),
                           nn.Linear(128,  128), nn.ReLU(),
                           nn.Linear(128,  c_out))
  
  transf1 = Node([in1.out0], GLOWCouplingBlock,
                 {'subnet_constructor':fc_constr},
                 name='Transform T_1')

• The RNVP and GLOW coupling blocks have an additional hyperparameter clamp. This is becuase, instead of the exponential function exp(s), we use exp( 2*c/pi * atan(x)) in the coupling blocks (clamp-parameter c). This leads to much more stable training and enables larger learning rates. Effecively, the mutliplication component of the coupling block is limited between exp(c) and 1/exp(c). The Jacobian determinant is thereby limited between ±D*c (dimensionaltiy of data D). In general, clamp = 2.0 is a good place to start:

  transf1 = Node([in1.out0], GLOWCouplingBlock,
                 {'subnet_constructor':fc_constr, 'clamp':2.0},
                 name='Transform T_1')

Examples

If you want full examples with training code etc., look through the experiments folder. The following only provides examples for constructing INNs by themselves.

# These imports and declarations apply to all examples
import torch.nn as nn

import FrEIA.framework as Ff
import FrEIA.modules as Fm

def subnet_fc(c_in, c_out):
    return nn.Sequential(nn.Linear(c_in, 512), nn.ReLU(),
                         nn.Linear(512,  c_out))

def subnet_conv(c_in, c_out):
    return nn.Sequential(nn.Conv2d(c_in, 256,   3, padding=1), nn.ReLU(),
                         nn.Conv2d(256,  c_out, 3, padding=1))

def subnet_conv_1x1(c_in, c_out):
    return nn.Sequential(nn.Conv2d(c_in, 256,   1), nn.ReLU(),
                         nn.Conv2d(256,  c_out, 1))

Simple GLOW-based INN

The following INN only has 2 input dimensions. It should be able to learn to generate most 2D distributions (gaussian mixtures, different shapes, ...), and can be easily visualized. Because of the 2D, it does not require permutations or orthogonal transforms between coupling blocks.

nodes = [Ff.InputNode(2, name='input')]

# Use a loop to produce a chain of coupling blocks
for k in range(8):
    nodes.append(Ff.Node(nodes[-1],
                         Fm.GLOWCouplingBlock,
                         {'subnet_constructor':subnet_fc, 'clamp':2.0},
                         name=F'coupling_{k}'))

nodes.append(Ff.OutputNode(nodes[-1], name='output'))
inn = Ff.ReversibleGraphNet(nodes)

Conditional INN

The following INN is able to perform conditional MNIST generation quite well. Note that is is not particularly efficient, with respect to the number of parameters. (See convolutional INN for that)

cond = Ff.ConditionNode(10, name='condition')
nodes = [Ff.InputNode(28*28, name='input')]

for k in range(12):
    nodes.append(Ff.Node(nodes[-1],
                         Fm.GLOWCouplingBlock,
                         {'subnet_constructor':subnet_fc, 'clamp':2.0},
                         conditions=cond,
                         name=F'coupling_{k}'))
    nodes.append(Ff.Node(nodes[-1],
                         Fm.PermuteRandom,
                         {'seed':k},
                         name=F'permute_{k}'))

nodes.append(Ff.OutputNode(nodes[-1], name='output'))
cinn = Ff.ReversibleGraphNet(nodes + [cond])

Convolutional INN

For the following architecture (which works e.g. for CIFAR10), 3/4 of the outputs are split off after some convolutions, which encode the local details, and the rest are transformed further to encode semantic content. This is important, because even for moderately sized images, it becomes infeasible to transform all dimenions through the full depth of the INN. Many dimensions will just enocde image noise, so we can split them off early.

nodes = [Ff.InputNode(3, 32, 32, name='input')]
ndim_x = 3 * 32 * 32

# Higher resolution convolutional part
for k in range(4):
   nodes.append(Ff.Node(nodes[-1],
                        Fm.GLOWCouplingBlock,
                        {'subnet_constructor':subnet_conv, 'clamp':1.2},
                        name=F'conv_high_res_{k}'))
   nodes.append(Ff.Node(nodes[-1],
                        Fm.PermuteRandom,
                        {'seed':k},
                        name=F'permute_high_res_{k}'))

nodes.append(Ff.Node(nodes[-1], Fm.IRevNetDownsampling, {}))

# Lower resolution convolutional part
for k in range(12):
   if k%2 == 0:
       subnet = subnet_conv_1x1
   else:
       subnet = subnet_conv

   nodes.append(Ff.Node(nodes[-1],
                        Fm.GLOWCouplingBlock,
                        {'subnet_constructor':subnet, 'clamp':1.2},
                        name=F'conv_low_res_{k}'))
   nodes.append(Ff.Node(nodes[-1],
                        Fm.PermuteRandom,
                        {'seed':k},
                        name=F'permute_low_res_{k}'))

# Make the outputs into a vector, then split off 1/4 of the outputs for the
# fully connected part
nodes.append(Ff.Node(nodes[-1], Fm.Flatten, {}, name='flatten'))
split_node = Ff.Node(nodes[-1],
                     Fm.Split1D,
                     {'split_size_or_sections':(ndim_x // 4, 3 * ndim_x // 4), 'dim':0},
                     name='split')
nodes.append(split_node)

# Fully connected part
for k in range(12):
   nodes.append(Ff.Node(nodes[-1],
                        Fm.GLOWCouplingBlock,
                        {'subnet_constructor':subnet_fc, 'clamp':2.0},
                        name=F'fully_connected_{k}'))
   nodes.append(Ff.Node(nodes[-1],
                        Fm.PermuteRandom,
                        {'seed':k},
                        name=F'permute_{k}'))

# Concatenate the fully connected part and the skip connection to get a single output
nodes.append(Ff.Node([nodes[-1].out0, split_node.out1],
                     Fm.Concat1d, {'dim':0}, name='concat'))
nodes.append(Ff.OutputNode(nodes[-1], name='output'))

conv_inn = Ff.ReversibleGraphNet(nodes)

Useful Tips & Engineering Heuristics

• Stochastic gradient descent will not work (well) for INNs. Use e.g. Adam instead.
• Gradient clipping can be useful if you are experiencing training instabilities, e.g. use torch.nn.utils.clip_grad_norm_
• Add some slight noise to the inputs (order of 1E-2). This stabilizes training and prevents sparse gradients, if there are some quantized or perfectly correlated input dimenions

For coupling blocks in particular:

• Use Xavier initialization for the weights. This prevents unstable training at the start.
• If your network is very deep (>30 coupling blocks), initialize the last layer in the subnetworks to zero. This means the INN as a whole is initialized to the identity, and you will not get NaNs at the first iteration.
• Do not forget permutations/orthogonal transforms between coupling blocks.
• Keep the subnetworks shallow (2-3 layers only), but wide (>= 128 neurons/ >= 64 conv. channels)
• Keep in mind that one coupling block contains between 4 and 12 individual convolutions or fully connected layers. So you may not have to use as many as you think, else the number of parameters will be huge.
• This being said, as the coupling blocks initialize to roughly the identity transform, it is hard to have too many coupling blocks and break the training completely (as opposed to a standard feed-forward NN).

For convolutional INNs in particular:

• Perform some kind of reshaping early, so the INN has >3 channels to work with
• Coupling blocks using 1x1 convolutions in the subnets seem important for the quality, they should constitute every other, or every third coupling block