Independent component analysis is a core framework within blind source separation for recovering latent source signals from observed mixtures under statistical independence assumptions. In this work, we propose PDGMM-VAE, a source-oriented variational autoencoder in which each latent dimension, interpreted explicitly as an individual source component, is assigned its own adaptive Gaussian mixture model prior. The proposed framework imposes heterogeneous per-dimension prior constraints, enabling different latent dimensions to model different non-Gaussian source marginals within a unified probabilistic encoder-decoder architecture. The parameters of these source-specific GMM priors are not fixed in advance, but are jointly learned together with the encoder and decoder under the overall training objective. Beyond the model construction itself, we provide a theoretical analysis clarifying why adaptive per-dimension prior design is meaningful in this setting. In particular, we show that heterogeneous per-dimension priors reduce latent permutation symmetry relative to homogeneous shared priors, and we further show that the KL regularization induced by the adaptive GMM prior creates source-specific attraction behavior that helps explain source-wise specialization during training. We also clarify the relation of the proposed model to the standard VAE and provide a weak recovery statement in an idealized linear low-noise regime. Experimental results on both linear and nonlinear mixing problems show that PDGMM-VAE can recover latent source signals and fit source-specific non-Gaussian marginals effectively. These results suggest that adaptive per-dimension mixture-prior design provides a principled and promising direction for VAE-based ICA and source-oriented generative modeling.
Figures & tables
Figure 1: Illustration of the proposed PDGMM-VAE framework. Each latent dimension is regularized by an adaptive Gaussian mixture prior whose parameters are jointly learned during training.
Figure 2: Training curves in the linear ICA experiment, including the total loss, posterior variances, GMM means, GMM variances, GMM weights, and per-source maximum correlations.
Figure 3: Comparison between the true sources and the inferred posterior means in the linear ICA experiment after z-score normalization. The shaded regions denote the posterior uncertainty bands.
Figure 4: True and estimated source distributions for Source 1 in the linear ICA experiment.
Figure 5: True and estimated source distributions for Source 2 in the linear ICA experiment.
Figure 6: True and estimated source distributions for Source 3 in the linear ICA experiment.
Figure 7: Training curves in the nonlinear ICA experiment, including the total loss, posterior variances, GMM means, GMM variances, GMM weights, and per-source maximum correlations.
Figure 8: Comparison between the true sources and the inferred posterior means in the nonlinear ICA experiment after z-score normalization. The shaded regions denote the posterior uncertainty bands.
Figure 9: True and estimated source distributions for Source 1 in the nonlinear ICA experiment.
Figure 10: True and estimated source distributions for Source 2 in the nonlinear ICA experiment.
Figure 11: True and estimated source distributions for Source 3 in the nonlinear ICA experiment.
We propose SAHMM-VAE, a source-wise adaptive Hidden Markov prior variational autoencoder for unsupervised blind source separation. Instead of treating the latent prior as a single generic regularizer, the proposed framework assigns each latent dimension its own adaptive regime-switching prior, so that different latent dimensions are pulled toward different source-specific temporal organizations during training. Under this formulation, source separation is not implemented as an external post-processing step; it is embedded directly into variational learning itself. The encoder, decoder, posterior parameters, and source-wise prior parameters are optimized jointly, where the encoder progressively learns an inference map that behaves like an approximate inverse of the mixing transformation, while the decoder plays the role of the generative mixing model. Through this coupled optimization, the gradual alignment between posterior source trajectories and heterogeneous HMM priors becomes the mechanism through which different latent dimensions separate into different source components. To instantiate this idea, we develop three branches within one common framework: a Gaussian-emission HMM prior, a Markov-switching autoregressive HMM prior, and an HMM state-flow prior with state-wise autoregressive flow transformations. Experiments show that the proposed framework achieves unsupervised source recovery while also learning meaningful source-wise switching structures. More broadly, the method extends our structured-prior VAE line from smooth, mixture-based, and flow-based latent priors to adaptive switching priors, and provides a useful basis for future work on interpretable and potentially identifiable latent source modeling.
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.
Yuan-Hao Wei, Fu-Hao Deng, Lin-Yong Cui +1
Hong Kong Polytechnic University · Wuhan University of Technology · The University of Hong Kong
We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on Rd, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.
Patrick Forré
AI4Science Lab Korteweg-de Vries Institute for Mathematics University of Amsterdam