Identifiable VAE

August 12, 2024

Identifiable VAE (iVAE) [1] is a variant of the Variational Autoencoder (VAE) model with the primary goal of making this model “identifiable.” This means that iVAE ensures that the latent variables \(z\) it learns are unique (up to permutation), unaffected by unwanted transformations such as permutation or nonlinear transformation. Below is a detailed explanation of the structure and functioning of iVAE.

1. Basics of Identifiability in Machine Learning #

• Identifiability in machine learning refers to the ability to uniquely determine the latent variables \(z\) from the observed data \(x\).
• In traditional VAE models, the latent space often lacks strong identifiability, meaning that the same data can lead to multiple different latent representations, making it difficult to interpret and apply in practice.
• iVAE is designed to address this issue by ensuring that the latent variables learned by the model are unique, corresponding to the true underlying sources in the data.

2. Structure of Identifiable VAE (iVAE) #

iVAE builds upon the structure of traditional VAE but with enhancements and additional conditions to ensure identifiability.

Encoder and Decoder #

• Encoder:

  • Maps the input data \(x\) to the latent space \(z\), similar to the traditional VAE.
  • However, in iVAE, the encoder is designed to take additional conditioning information \(u\), which could be external variables or other features of the data that help distinguish between different latent sources.
  • Formula: \( q_{\phi}(z|x, u) \), where \(u\) is the conditioning variable.

• Decoder:

  • Receives the latent variable \(z\) from the encoder and reconstructs the data \(x'\).
  • Like in VAE, the decoder in iVAE is optimized to reconstruct the input data from the latent space, but now with an additional conditioning factor.
  • Formula: \( p_{\theta}(x|z, u) \).

Conditioning the Latent Space #

• Latent Space:
  • To ensure identifiability, iVAE conditions the latent space \(z\) on the variables \(u\).
  • This conditioning ensures that the same latent source across different datasets can be uniquely identified, preventing unwanted permutations of latent variables.

Conditional Posterior Distribution #

• iVAE extends the posterior distribution by conditioning on \(u\), rather than simply using \(q(z|x)\) as in traditional VAE.
• This means that when learning latent variables, iVAE uses additional information \(u\) to distinguish between different latent sources, making them identifiable.

3. Objective Function #

• The objective function of iVAE is similar to VAE but includes additional terms to ensure identifiability: \[ \mathcal{L}(\theta, \phi; x, u) = \mathbb{E}_{q_{\phi}(z|x,u)}[\log p_{\theta}(x|z,u)] - D_{\text{KL}}(q_{\phi}(z|x,u) \| p(z|u)) \]
  • Term 1: Reconstruct the data \(x\) from \(z\) and \(u\), similar to traditional VAE.
  • Term 2: KL-divergence between the conditional posterior distribution and the conditional prior distribution \(p(z|u)\), which helps condition and maintain the identifiability of the latent space.

4. Practical Applications #

iVAE can be applied in many fields where identifying latent sources is crucial:

• Source Separation: In applications like audio source separation or brain signal separation, iVAE helps ensure that the latent sources learned from the mixed signal are unique and identifiable.
• Latent Factor Analysis: iVAE is useful for analyzing latent factors in complex data like medical or financial data, where accurately identifying these factors is essential.
• Interpretable Representation Learning: iVAE helps in learning latent representations that are interpretable, aiding in decision-making and data analysis.

Conclusion #

Identifiable VAE (iVAE) is a powerful variant of VAE that addresses the identifiability issue in latent space by conditioning on external variables. This makes iVAE a valuable tool in applications where accurately identifying latent sources or factors in the data is important, and it enhances the interpretability and reliability of machine learning models.

Tài Liệu Tham Khảo #

[1] I. Khemakhem, D. P. Kingma, R. P. Monti, and A. Hyvärinen, “Variational Autoencoders and Nonlinear ICA: A Unifying Framework,” Dec. 21, 2020, arXiv: arXiv:1907.04809. doi: 10.48550/arXiv.1907.04809.

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