Yan-Jie Sun

Yuan-Hao Wei, Yan-Jie Sun, Chen Zhang

Inference and inverse problems are closely related concepts, both fundamentally involving the deduction of unknown causes or parameters from observed data. Bayesian inference, a powerful class of methods, is often employed to solve a variety of problems, including those related to causal inference. Variational inference, a subset of Bayesian inference, is primarily used to efficiently approximate complex posterior distributions. Variational Autoencoders (VAEs), which combine variational inference with deep learning, have become widely applied across various domains. This study explores the potential of VAEs for solving inverse problems, such as Independent Component Analysis (ICA), without relying on an explicit inverse mapping process. Unlike other VAE-based ICA methods, this approach discards the encoder in the VAE architecture, directly setting the latent variables as trainable parameters. In other words, the latent variables are no longer outputs of the encoder but are instead optimized directly through the objective function to converge to appropriate values. We find that, with a suitable prior setup, the latent variables, represented by trainable parameters, can exhibit mutually independent properties as the parameters converge, all without the need for an encoding process. This approach, referred to as the Half-VAE, bypasses the inverse mapping process by eliminating the encoder. This study demonstrates the feasibility of using the Half-VAE to solve ICA without the need for an explicit inverse mapping process.

Yuan-Hao Wei, Yan-Jie Sun

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 signal, is assigned its own Gaussian mixture model prior. Unlike conventional VAE formulations with a shared simple prior, the proposed framework imposes per-dimension heterogeneous prior constraints, enabling the model to capture diverse non-Gaussian source statistics and thereby promote source separation under a probabilistic encoder-decoder architecture. Importantly, the parameters of these per-dimension GMM priors are not fixed in advance, but are adaptively learned and automatically refined toward convergence together with the encoder and decoder parameters under the overall training objective. Within this formulation, the encoder serves as a demixing mapping from observations to latent sources, while the decoder reconstructs the observed mixtures from the inferred components. The proposed model provides a systematic study of an idea that had previously only been noted in our preliminary form, namely, equipping different latent sources with different GMM priors for ICA, and formulates it as a full VAE framework with end-to-end training and per-dimension prior learning. Experimental results on both linear and nonlinear mixing problems demonstrate that PDGMM-VAE can recover latent source signals and achieve satisfactory separation performance.

Yuan-Hao Wei, Fu-Hao Deng, Lin-Yong Cui et al.

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, Yan-Jie Sun

This study advances the Variational Autoencoder (VAE) framework by addressing challenges in Independent Component Analysis (ICA) under both determined and underdetermined conditions, focusing on enhancing the independence and interpretability of latent variables. Traditional VAEs map observed data to latent variables and back via an encoder-decoder architecture, but struggle with underdetermined ICA where the number of latent variables exceeds observed signals. The proposed Half Adversarial VAE (Half-AVAE) builds on the encoder-free Half-VAE framework, eliminating explicit inverse mapping to tackle underdetermined scenarios. By integrating adversarial networks and External Enhancement (EE) terms, Half-AVAE promotes mutual independence among latent dimensions, achieving factorized and interpretable representations. Experiments with synthetic signals demonstrate that Half-AVAE outperforms baseline models, including GP-AVAE and Half-VAE, in recovering independent components under underdetermined conditions, as evidenced by lower root mean square errors. The study highlights the flexibility of VAEs in variational inference, showing that encoder omission, combined with adversarial training and structured priors, enables effective solutions for complex ICA tasks, advancing applications in disentanglement, causal inference, and generative modeling.

Yuan-Hao Wei, Fu-Hao Deng, Lin-Yong Cui et al.

The interpretability of generative models is considered a key factor in demonstrating their effectiveness and controllability. The generated data are believed to be determined by latent variables that are not directly observable. Therefore, disentangling, decoupling, decomposing, causal inference, or performing Independent Component Analysis (ICA) in the latent variable space helps uncover the independent factors that influence the attributes or features affecting the generated outputs, thereby enhancing the interpretability of generative models. As a generative model, Variational Autoencoders (VAEs) combine with variational Bayesian inference algorithms. Using VAEs, the inverse process of ICA can be equivalently framed as a variational inference process. In some studies, Gaussian processes (GPs) have been introduced as priors for each dimension of latent variables in VAEs, structuring and separating each dimension from temporal or spatial perspectives, and encouraging different dimensions to control various attributes of the generated data. However, GPs impose a significant computational burden, resulting in substantial resource consumption when handling large datasets. Essentially, GPs model different temporal or spatial structures through various kernel functions. Structuring the priors of latent variables via kernel functions-so that different kernel functions model the correlations among sequence points within different latent dimensions-is at the core of achieving disentanglement in VAEs. The proposed Structured Kernel Regression VAE (SKR-VAE) leverages this core idea in a more efficient way, avoiding the costly kernel matrix inversion required in GPs. This research demonstrates that, while maintaining ICA performance, SKR-VAE achieves greater computational efficiency and significantly reduced computational burden compared to GP-VAE.