Poisson Identifiable VAE (pi-VAE)

This is a Pytorch implementation of Poisson Identifiable VAE (pi-VAE), used to construct latent variable models of neural activity while simultaneously modeling the relation between the latent and task variables (non-neural variables, e.g. sensory, motor, and other externally observable states).

The original implementation by Ding Zhou and Xue-Xin Wei in Tensorflow 1.13 is available here.

Another Pytorch implementation by Lyndon Duong is available here.

A special thank you to Zhongxuan Wu who helped in the design and testing of this implementation.

Install

pip install pi-vae-pytorch

From Source

For those interested in modifying and testing the codebase, using an editable pip installation is recommended:

# pi-vae-pytorch/

pip install -e .

Model Architecture

pi-VAE is comprised of three main components: the encoder, the label prior estimator, and the decoder.

MLP Structure

The Multi Layer Perceptron (MLP) is the primary building block of the aforementioned components. Each MLP used in this implementation is configurable by specifying the appropriate parameters when PiVAE is initialized:

• number of hidden layers
• hidden layer dimension
  • applied to all hidden layers within a given MLP
• hidden layer activation function
  • applied to all non-output layer activations within a given MLP

Encoder

The model's encoder is comprised of a single MLP, which learns the distribution q(z | x).

Label Prior Estimator

The model's label prior estimator learns the distribution p(z | u). In the discrete label regime this module is comprised of a single nn.Embedding layer, while in the continuous label regime the module is comprised of a MLP.

Decoder

The model's decoder learns to map a latent sample z to the predicted firing rate of the latent sample. Inputs to the decoder are passed through the following submodules.

NFlowLayer

This module is comprised of a MLP which maps z to the concatenation of z and t(z).

GINBlock

Outputs from the NFlowLayer are passed to a series of GINBlock modules. Each GINBlock is comprised of a specified number of AffineCouplingLayer modules. Each AffineCouplingLayer is comprised of a MLP and performs an affine coupling transformation.

Usage

import torch
from pi_vae_pytorch import PiVAE

model = PiVAE(
    x_dim = 100,
    u_dim = 3,
    z_dim = 2,
    discrete_labels=False
)

x = torch.randn(1, 100) # Size([n_samples, x_dim])

u = torch.randn(1, 3) # Size([n_samples, u_dim])

outputs = model(x, u) # Dict

Parameters

• x_dim: int
  Dimension of observation x

• u_dim: int
  Dimension of label u

• z_dim: int
  Dimension of latent z

• discrete_labels: bool

  • Default: True

  Flag denoting u's label type - True: discrete or False: continuous.

• encoder_n_hidden_layers: int

  • Default: 2

  Number of hidden layers in the MLP of the model's encoder.

• encoder_hidden_layer_dim: int

  • Default: 120

  Dimensionality of each hidden layer in the MLP of the model's encoder.

• encoder_hidden_layer_activation: nn.Module

  • Default: nn.Tanh

  Activation function applied to the outputs of each hidden layer in the MLP of the model's encoder.

• decoder_n_gin_blocks: int

  • Default: 2

  Number of GIN blocks used within the model's decoder.

• decoder_gin_block_depth: int

  • Default: 2

  Number of AffineCouplingLayers which comprise each GIN block.

• decoder_affine_input_layer_slice_dim: int

  • Default None (corresponds to x_dim // 2)

  Index at which to split an n-dimensional input x.

• decoder_affine_n_hidden_layers: int

  • Default: 2

  Number of hidden layers in the MLP of the model's encoder.

• decoder_affine_hidden_layer_dim: int

  • Default: None (corresponds to x_dim // 4)

  Dimensionality of each hidden layer in the MLP of each AffineCouplingLayer.

• decoder_affine_hidden_layer_activation: nn.Module

  • Default: nn.ReLU

  Activation function applied to the outputs of each hidden layer in the MLP of each AffineCouplingLayer.

• decoder_nflow_n_hidden_layers: int

  • Default: 2

  Number of hidden layers in the MLP of the decoder's NFlowLayer.

• decoder_nflow_hidden_layer_dim: int

  • Default: None (corresponds to x_dim // 4)

  Dimensionality of each hidden layer in the MLP of the decoder's NFlowLayer.

• decoder_nflow_hidden_layer_activation: nn.Module

  • Default: nn.ReLU

  Activation function applied to the outputs of each hidden layer in the MLP of the decoder's NFlowLayer.

• decoder_observation_model: str

  • Default: poisson
  • One of gaussian or poisson

  Observation model used by the model's decoder.

• decoder_fr_clamp_min: float

  • Default: 1E-7
  • Only applied when decoder_observation_model="poisson"

  Mininimum threshold used when clamping decoded firing rates.

• decoder_fr_clamp_max: float

  • Default: 1E7
  • Only applied when decoder_observation_model="poisson"

  Maximum threshold used when clamping decoded firing rates.

• z_prior_n_hidden_layers: int

  • Default: 2
  • Only applied when discrete_labels=False

  Number of hidden layers in the MLP of the label prior estimator module.

• z_prior_hidden_layer_dim: int

  • Default: 20
  • Only applied when discrete_labels=False

  Dimensionality of each hidden layer in the MLP of the label prior estimator module.

• z_prior_hidden_layer_activation: nn.Module

  • Default: nn.Tanh
  • Only applied when discrete_labels=False

  Activation function applied to the outputs of each hidden layer in the MLP of the label prior estimator module.

Returns

A Dict with the following items.

• firing_rate: Tensor

  • Size([n_samples, x_dim])

  Predicted firing rates of z_sample.

• lambda_mean: Tensor

  • Size([n_samples, z_dim])

  Mean for each sample using label prior p(z | u).

• lambda_log_variance: Tensor

  • Size([n_samples, z_dim])

  Log of variance for each sample using label prior p(z | u).

• posterior_mean: Tensor

  • Size([n_samples, z_dim])

  Mean for each sample using full posterior of q(z | x,u) ~ q(z | x) × p(z | u).

• posterior_log_variance: Tensor

  • Size([n_samples, z_dim])

  Log of variance for each sample using full posterior of q(z | x,u) ~ q(z | x) × p(z | u).

• z_mean: Tensor

  • Size([n_samples, z_dim])

  Mean for each sample using approximation of q(z | x).

• z_log_variance: Tensor

  • Size([n_samples, z_dim])

  Log of variance for each sample using approximation of q(z | x).

• z_sample: Tensor

  • Size([n_samples, z_dim])

  Generated latents z.

Loss Function

pi-VAE learns the deep generative model and the approximate posterior q(z | x, u) of the true posterior p(z | x, u) by maximizing the evidence lower bound (ELBO) of p(x | u).

Poisson observation model

from pi_vae_pytorch.utils import compute_loss

outputs = model(x, u) # Initialized with decoder_observation_model="poisson" 

loss = compute_loss(
    x=x,
    firing_rate=outputs["firing_rate"],
    lambda_mean=outputs["lambda_mean"],
    lambda_log_variance=outputs["lambda_log_variance"],
    posterior_mean=outputs["posterior_mean"],
    posterior_log_variance=outputs["posterior_log_variance"],
    observation_model=model.decoder_observation_model
)

loss.backward()

Gaussian observation model

from pi_vae_pytorch.utils import compute_loss

outputs = model(x, u) # Initialized with decoder_observation_model="gaussian" 

loss = compute_loss(
    x=x,
    firing_rate=outputs["firing_rate"],
    lambda_mean=outputs["lambda_mean"],
    lambda_log_variance=outputs["lambda_log_variance"],
    posterior_mean=outputs["posterior_mean"],
    posterior_log_variance=outputs["posterior_log_variance"],
    observation_model=model.decoder_observation_model,
    observation_noise_model=model.observation_noise_model
)

loss.backward()

Parameters

• x: Tensor

  • Size([n_samples, x_dim])

  Observations x.

• firing_rate: Tensor

  • Size([n_samples, x_dim])

  Predicted firing rate of generated latent z.

• lambda_mean: Tensor

  • Size([n_samples, z_dim])

  Means from label prior p(z | u).

• lambda_log_variance: Tensor

  • Size([n_samples, z_dim])

  Log of variances from label prior p(z | u).

• posterior_mean: Tensor

  • Size([n_samples. z_dim])

  Means from full posterior of q(z | x,u) ~ q(z | x) × p(z | u).

• posterior_log_variance: Tensor

  • Size([n_samples. z_dim])

  Log of variances from full posterior of q(z | x,u) ~ q(z | x) × p(z | u).

• observation_model: str

  • One of poisson or gaussian
  • Should use the same value passed to decoder_observation_model when initializing PiVAE.

  The observation model used by pi-VAE's decoder.

• observation_noise_model: nn.Module

  • Default: None
  • Only applied when observation model="gaussian"

  The noise model used when pi-VAE's decoder utilizes a Gaussian observation model. When PiVAE is initialized with decoder_observation_model="gaussian", the model's observation_noise_model attribute should be used.

Citation

@misc{zhou2020learning,
    title={Learning identifiable and interpretable latent models of high-dimensional neural activity using pi-VAE}, 
    author={Ding Zhou and Xue-Xin Wei},
    year={2020},
    eprint={2011.04798},
    archivePrefix={arXiv},
    primaryClass={stat.ML}
}