Model

Model initialization and architecture.

The architecture of the interpretable autoregressive \(\beta\)-VAE works in the following manner: Given the displacements \(\mathbf{\Delta x}(t)\) of a diffusion trajectory, the encoder (orange) compresses them into an interpretable latent space (blue), in which few neurons (dark blue) represent physical features of the input data while others are noised out (light blue). An autoregressive decoder (green) generates from this latent representation the displacements \(\mathbf{\Delta x}'(t)\) of a new trajectory recursively, considering a certain receptive field RF (light green cone).

Initialization

As the architecture can be quite deep, a careful initialization is needed (see weight_init function in the model class). We initialize the weights with normal Kaiming init in fan_out mode, taking into account that we use the nonlinear activation function ReLU.

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init_cnn

 init_cnn (m)

Initialize weights with kaiming normal in fan_out mode and bias to 0

VAE

We implement a 1D convolutional variational autoencoder.

The latent neurons are probabilistic, i.e., they are sampled following a distribution. The reparameterization trick provides the means to allow backpropagation by externalizing the sampling noise.

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reparameterize

 reparameterize (mu, logvar)

Samples from a normal distribution using the reparameterization trick.

	Type	Details
mu	torch.Tensor	Mean of the normal distribution. Shape (batch_size, latent_dim)
logvar	torch.Tensor	Diagonal log variance of the normal distribution. Shape (batch_size, latent_dim)
Returns	torch.Tensor	Sampled latent z tensor as \(z=\epsilon\sigma+\mu\)

We also take into account the sizes after n convolutions are applied to automate the model construction.

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output_size_after_n_convt

 output_size_after_n_convt (n, input_size, kernel_size, stride=1,
                            padding=0, output_padding=0, dilation=1)

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output_size_convt

 output_size_convt (input_size, kernel_size, stride=1, padding=0,
                    output_padding=0, dilation=1)

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output_size_after_n_conv

 output_size_after_n_conv (n, input_size, kernel_size, stride=1,
                           padding=0, dilation=1)

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output_size_conv

 output_size_conv (input_size, kernel_size, stride=1, padding=0,
                   dilation=1)

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View

 View (size)

Use as (un)flattening layer

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VAEConv1d

 VAEConv1d (nf, encoder, decoder, o_dim:int, nc_in=1, nc_out=6, z_dim=6,
            beta=0, avg_size=24, **kwargs)

1-dimensional convolutional VAE architecture

	Type	Default	Details
nf			number of filters
encoder			list of Encoder’s dense layers sizes
decoder			list of Decoder’s dense layers sizes
o_dim	int		input size (T)
nc_in	int	1	number of input channels
nc_out	int	6	number of output channels
z_dim	int	6	number of latent neurons
beta	int	0	weight of the KLD loss
avg_size	int	24	output size of the pooling layers
kwargs

VAE + WaveNet

We implement an extensible version of a VAE with WaveNet as the autoregressive decoder.

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sample_from_mix_gaussian

 sample_from_mix_gaussian (y, log_scale_min=-12.0)

Sample from (discretized) mixture of gaussian distributions

	Type	Default	Details
y	Tensor		Mixture of Gaussians parameters. Shape (B x C x T)
log_scale_min	float	-12.0	Log scale minimum value
Returns	Tensor

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DilatedCausalConv1d

 DilatedCausalConv1d (mask_type, in_channels, out_channels, kernel_size=2,
                      dilation=1, bias=True, use_pad=True)

Dilated causal convolution for WaveNet

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ResidualBlock

 ResidualBlock (res_channels, skip_channels, kernel_size, dilation,
                c_channels=0, g_channels=0, bias=True, use_pad=True)

Residual block with conditions and gate mechanism

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VAEWaveNet

 VAEWaveNet (in_channels=1, res_channels=16, skip_channels=16,
             c_channels=6, g_channels=0, out_channels=1,
             res_kernel_size=3, layer_size=4, stack_size=1,
             out_distribution='normal', discrete_channels=256,
             num_mixtures=1, use_pad=False, weight_norm=False, **kwargs)

VAE with autoregressive decoder

	Type	Default	Details
in_channels	int	1	input channels
res_channels	int	16	residual channels
skip_channels	int	16	skip connections channels
c_channels	int	6	local conditioning
g_channels	int	0	global conditioning
out_channels	int	1	output channels
res_kernel_size	int	3	kernel_size of residual blocks dilated layers
layer_size	int	4	Largest dilation is 2^layer_size
stack_size	int	1	number of layers stacks
out_distribution	str	normal
discrete_channels	int	256
num_mixtures	int	1
use_pad	bool	False
weight_norm	bool	False
kwargs

We can create a model by specifying its parameters in a dict.

model_args = dict(# VAE #########################
                  o_dim=400,
                  nc_in=1, nc_out=6,
                  nf=[16]*4,
                  avg_size=16,
                  encoder=[200,100],
                  z_dim=6,
                  decoder=[100,200],
                  beta=0,
                  # WaveNet ########
                  in_channels=1,
                  res_channels=16,skip_channels=16,
                  c_channels=6,
                  g_channels=0,
                  res_kernel_size=3,
                  layer_size=4,  # 6
                  stack_size=1,
                  out_distribution= "Normal",
                  num_mixtures=1,
                  use_pad=False,
                  model_name = 'SPIVAE',
                 )
model = VAEWaveNet(**model_args)

Printing the model object will reveal the declared layers.

model

VAEWaveNet(
  (vae): VAEConv1d(
    (encoder): Sequential(
      (0): Conv1d(1, 16, kernel_size=(3,), stride=(1,))
      (1): ReLU(inplace=True)
      (2): Conv1d(16, 16, kernel_size=(3,), stride=(1,))
      (3): ReLU(inplace=True)
      (4): Conv1d(16, 16, kernel_size=(3,), stride=(1,))
      (5): ReLU(inplace=True)
      (6): Conv1d(16, 16, kernel_size=(3,), stride=(1,))
      (7): ReLU(inplace=True)
      (8): AdaptiveConcatPool1d(
        (ap): AdaptiveAvgPool1d(output_size=16)
        (mp): AdaptiveMaxPool1d(output_size=16)
      )
      (9): View()
      (10): Linear(in_features=512, out_features=200, bias=True)
      (11): ReLU(inplace=True)
      (12): Linear(in_features=200, out_features=100, bias=True)
      (13): ReLU(inplace=True)
      (14): Linear(in_features=100, out_features=12, bias=True)
    )
    (decoder): Sequential(
      (0): Linear(in_features=6, out_features=100, bias=True)
      (1): ReLU(inplace=True)
      (2): Linear(in_features=100, out_features=200, bias=True)
      (3): ReLU(inplace=True)
      (4): Linear(in_features=200, out_features=512, bias=True)
      (5): ReLU(inplace=True)
      (6): View()
    )
    (convt): Sequential(
      (0): ConvTranspose1d(16, 16, kernel_size=(3,), stride=(1,))
      (1): ReLU(inplace=True)
      (2): ConvTranspose1d(16, 16, kernel_size=(3,), stride=(1,))
      (3): ReLU(inplace=True)
      (4): ConvTranspose1d(16, 16, kernel_size=(3,), stride=(1,))
      (5): ReLU(inplace=True)
      (6): ConvTranspose1d(16, 6, kernel_size=(3,), stride=(1,))
      (7): ReLU(inplace=True)
    )
  )
  (init_conv): Conv1d(1, 16, kernel_size=(1,), stride=(1,))
  (causal): DilatedCausalConv1d(
    (conv): Conv1d(16, 16, kernel_size=(2,), stride=(1,))
  )
  (res_stack): ModuleList(
    (0): ResidualBlock(
      (dilated): DilatedCausalConv1d(
        (conv): Conv1d(16, 32, kernel_size=(3,), stride=(1,))
      )
      (conv_c): Conv1d(6, 32, kernel_size=(1,), stride=(1,), bias=False)
      (conv_res): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
      (conv_skip): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
    )
    (1): ResidualBlock(
      (dilated): DilatedCausalConv1d(
        (conv): Conv1d(16, 32, kernel_size=(3,), stride=(1,), dilation=(2,))
      )
      (conv_c): Conv1d(6, 32, kernel_size=(1,), stride=(1,), bias=False)
      (conv_res): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
      (conv_skip): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
    )
    (2): ResidualBlock(
      (dilated): DilatedCausalConv1d(
        (conv): Conv1d(16, 32, kernel_size=(3,), stride=(1,), dilation=(4,))
      )
      (conv_c): Conv1d(6, 32, kernel_size=(1,), stride=(1,), bias=False)
      (conv_res): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
      (conv_skip): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
    )
    (3): ResidualBlock(
      (dilated): DilatedCausalConv1d(
        (conv): Conv1d(16, 32, kernel_size=(3,), stride=(1,), dilation=(8,))
      )
      (conv_c): Conv1d(6, 32, kernel_size=(1,), stride=(1,), bias=False)
      (conv_res): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
      (conv_skip): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
    )
  )
  (out_conv): Sequential(
    (0): ReLU(inplace=True)
    (1): Conv1d(16, 16, kernel_size=(1,), stride=(1,))
    (2): ReLU(inplace=True)
    (3): Conv1d(16, 9, kernel_size=(1,), stride=(1,))
  )
)

Training example

With the data and the model, we can already start training.

DEVICE= 'cpu' # 'cuda'
print(DEVICE)

cpu

Ds = np.linspace(2e-5,2e-2,10)
alphas = np.linspace(0.2,1.8,21)
n_alphas,n_Ds = len(alphas), len(Ds)
ds_args = dict(path="../../data/raw/", model='fbm', # 'sbm'
               N=int(6_000/n_alphas/n_Ds*2),
               T=400,
               D=Ds, alpha=alphas,seed=0,
               valid_pct=0.2,
               bs=2**8,
               N_save=6_000, T_save=400,
              )
model_args = dict(# VAE ###########################
                  o_dim=ds_args['T']-1,
                  nc_in=1, nc_out=6,
                  nf=[16]*4,
                  avg_size=16,
                  encoder=[200,100],
                  z_dim=6,
                  decoder=[100,200],
                  beta=0,
                  # WaveNet ########
                  in_channels=1,
                  res_channels=16,skip_channels=16,
                  c_channels=6,
                  g_channels=0,
                  res_kernel_size=3,
                  layer_size=4,  # 6  # Largest dilation is 2**layer_size
                  stack_size=1,
                  out_distribution= "Normal",
                  num_mixtures=1,
                  use_pad=False,
                  model_name = 'SPIVAE',
                 )

dls = load_data(ds_args)

model = VAEWaveNet(**model_args).to(DEVICE)

loss_fn = Loss(model.receptive_field, model.c_channels, 
               beta=model_args['beta'], reduction='mean')

learn = Learner(dls, model, loss_func=loss_fn,)

learn.fit_one_cycle(4, lr_max=1e-4)

epoch	train_loss	valid_loss	time
0	0.982180	0.949590	00:28
1	0.934559	0.880010	00:25
2	0.882010	0.822553	00:26
3	0.843744	0.810182	00:29