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AR-Flow VAE: A Structured Autoregressive Flow Prior Variational Autoencoder for Unsupervised Blind Source Separation

Yuan-Hao Wei, Fu-Hao Deng, Lin-Yong Cui, Yan-Jie Sun

Abstract

Blind source separation (BSS) seeks to recover latent source signals from observed mixtures. Variational autoencoders (VAEs) offer a natural perspective for this problem: the latent variables can be interpreted as source components, the encoder can be viewed as a demixing mapping from observations to sources, and the decoder can be regarded as a remixing process from inferred sources back to observations. In this work, we propose AR-Flow VAE, a novel VAE-based framework for BSS in which each latent source is endowed with a parameter-adaptive autoregressive flow prior. This prior significantly enhances the flexibility of latent source modeling, enabling the framework to capture complex non-Gaussian behaviors and structured dependencies, such as temporal correlations, that are difficult to represent with conventional priors. In addition, the structured prior design assigns distinct priors to different latent dimensions, thereby encouraging the latent components to separate into different source signals under heterogeneous prior constraints. Experimental results validate the effectiveness of the proposed architecture for blind source separation. More importantly, this work provides a foundation for future investigations into the identifiability and interpretability of AR-Flow VAE.

AR-Flow VAE: A Structured Autoregressive Flow Prior Variational Autoencoder for Unsupervised Blind Source Separation

Abstract

Blind source separation (BSS) seeks to recover latent source signals from observed mixtures. Variational autoencoders (VAEs) offer a natural perspective for this problem: the latent variables can be interpreted as source components, the encoder can be viewed as a demixing mapping from observations to sources, and the decoder can be regarded as a remixing process from inferred sources back to observations. In this work, we propose AR-Flow VAE, a novel VAE-based framework for BSS in which each latent source is endowed with a parameter-adaptive autoregressive flow prior. This prior significantly enhances the flexibility of latent source modeling, enabling the framework to capture complex non-Gaussian behaviors and structured dependencies, such as temporal correlations, that are difficult to represent with conventional priors. In addition, the structured prior design assigns distinct priors to different latent dimensions, thereby encouraging the latent components to separate into different source signals under heterogeneous prior constraints. Experimental results validate the effectiveness of the proposed architecture for blind source separation. More importantly, this work provides a foundation for future investigations into the identifiability and interpretability of AR-Flow VAE.
Paper Structure (22 sections, 48 equations, 3 figures, 1 algorithm)

This paper contains 22 sections, 48 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structured Priors in VAEs
  3. Proposed Method: AR-Flow VAE
  4. Overview
  5. Generative Model
  6. Observation model.
  7. Variational Posterior
  8. Per-Source Autoregressive Flow Prior
  9. Initial state
  10. Autoregressive backbone
  11. Autoregressive flow transform
  12. Exact Prior Density by Change of Variables
  13. Step 1: Transformation from source trajectory to AR residual
  14. Step 2: Transformation from AR residual to flow noise
  15. Step 3: Base density evaluation
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Typical structured-prior VAE for BSS. The encoder is interpreted as a demixing operator from observed mixtures to latent source components, while the decoder acts as a generative mixing operator that reconstructs the observations from the inferred sources. Each latent dimension is assigned an individual parameterized prior, and the corresponding posterior is regularized toward that prior through the KL divergence.
  • Figure 2: Recovered sources in the synthetic experiment. For each source (Z-score normalized), the true signal, the inferred posterior mean, and the corresponding 95% confidence interval are shown.
  • Figure 3: Training traces of AR-Flow VAE in the synthetic experiment, including the total loss, overall maximum correlation, posterior variances, autoregressive parameters, innovation scales, and per-source best correlations.