# StrEBM: A Structured Latent Energy-Based Model for Blind Source Separation

arXiv: https://arxiv.org/abs/2604.17381 · HTML: https://mlsanity.com/paper/2604.17381/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 19 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [GP-VAE: Deep Probabilistic Time Series Imputation](https://mlsanity.com/paper/1907.04155/) (2019)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [Composing graphical models with neural networks for structured representations and fast inference](https://mlsanity.com/paper/1603.06277/) (2016)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [Training Deep Energy-Based Models with f-Divergence Minimization](https://mlsanity.com/paper/2003.03463/) (2020)
- [Revisiting Structured Variational Autoencoders](https://mlsanity.com/paper/2305.16543/) (2023)
- [PDGMM-VAE: A Variational Autoencoder with Adaptive Per-Dimension Gaussian Mixture Model Priors for Nonlinear ICA](https://mlsanity.com/paper/2603.23547/) (2026)
- [StrADiff: A Structured Source-Wise Adaptive Diffusion Framework for Linear and Nonlinear Blind Source Separation](https://mlsanity.com/paper/2604.04973/) (2026)
- [SAHMM-VAE: A Source-Wise Adaptive Hidden Markov Prior Variational Autoencoder for Unsupervised Blind Source Separation](https://mlsanity.com/paper/2603.25776/) (2026)

## Built on by (most cited first, 1 of 1)

- [StrTransformer: Source-Wise Structured Transformers for Unsupervised Blind Source Recovery](https://mlsanity.com/paper/2605.25648/) (2026)

## Abstract (arXiv)

This paper proposes StrEBM, a structured latent energy-based model for source-wise structured representation learning. The framework is motivated by a broader goal of promoting identifiable and decoupled latent organization by assigning different latent dimensions their own learnable structural biases, rather than constraining the entire latent representation with a single shared energy. In this sense, blind source separation is adopted here as a concrete and verifiable testbed, through which the evolution of latent dimensions toward distinct underlying components can be directly examined. In the proposed framework, latent trajectories are optimized directly together with an observation-generation map and source-wise structural parameters. Each latent dimension is associated with its own energy-based formulation, allowing different latent components to gradually evolve toward distinct source-like roles during training. In the present study, this source-wise energy design is instantiated using Gaussian-process-inspired energies with learnable length-scales, but the framework itself is not restricted to Gaussian processes and is intended as a more general structured latent EBM formulation. Experiments on synthetic multichannel signals under linear and nonlinear mixing settings show that the proposed model can recover source components effectively, providing an initial empirical validation of the framework. At the same time, the study reveals important optimization characteristics, including slow late-stage convergence and reduced stability under nonlinear observation mappings. These findings not only clarify the practical behavior of the current GP-based instantiation, but also establish a basis for future investigation of richer source-wise energy families and more robust nonlinear optimization strategies.