Unsupervised Disentanglement Without Compromises : How Functional Orthogonality Enforces Identifiability

Disentangled representation learning aims to map independent factors of variation to independent representation components. On one hand, purely unsupervised approaches have proven successful on fully disentangled synthetic data, but fail to recover semantic factors from real data without strong inductive biases. On the other hand, supervised approaches are unstable and hard to scale to large attribute sets because they rely on adversarial objectives or auxiliary classifiers. We introduce \textsc{XFactors}, a weakly-supervised VAE framework that disentangles and provides explicit control over a chosen set of factors. Building on the Disentangled Information Bottleneck perspective, we decompose the representation into a residual subspace $\mathcal{S}$ and factor-specific subspaces $\mathcal{T}_1,\ldots,\mathcal{T}_K$ and a residual subspace $\mathcal{S}$. Each target factor is encoded in its assigned $\mathcal{T}_i$ through contrastive supervision: an InfoNCE loss pulls together latents sharing the same factor value and pushes apart mismatched pairs. In parallel, KL regularization imposes a Gaussian structure on both $\mathcal{S}$ and the aggregated factor subspaces, organizing the geometry without additional supervision for non-targeted factors and avoiding adversarial training and classifiers. Across multiple datasets, with constant hyperparameters, \textsc{XFactors} achieves state-of-the-art disentanglement scores and yields consistent qualitative factor alignment in the corresponding subspaces, enabling controlled factor swapping via latent replacement. We further demonstrate that our method scales correctly with increasing latent capacity and evaluate it on the real-world dataset CelebA. Our code is available at \href{https://github.com/ICML26-anon/XFactors}{github.com/ICML26-anon/XFactors}.

There is a gap between the theoretical foundations of disentanglement and the practice of modern representation learning. Existing theoretical frameworks, particularly Independent Component Analysis (ICA) and its nonlinear variants, assume a generative model with statistically independent latent variables underlying the data so that disentanglement amounts to identifying the latents that could have generated the data. This generative framework is interpretable and theoretically justified, but its strong assumptions make it difficult to apply to modern representation learning. Modern pretrained encoders often learn features that exhibit disentangled properties without making generative assumptions, yet there is no general theory for interpreting these features as independent factors of variation. We take a step toward such a theory by introducing Riemannian ICA (RICA), which replaces ICA's global generative model with local geometric structure. RICA is founded on the observation that in ICA, the factors of variation underlying a data point can be understood through radial curves emanating from the point that map to axis-aligned lines in the latent space. We formalize this perspective using Riemannian geometry and introduce our theory in a way that is consistent with the existing generative approach. Our main contribution is the disentanglement tensor, which encodes a second-order notion of disentanglement that we call pointwise disentanglement. This tensor depends on the Hessian of the data log likelihood as well as the Ricci curvature induced by the model. In a controlled source recovery setting with known ground-truth sources, RICA recovers sources across several manifolds, while the success of ICA baselines depends on the coordinates used to represent the observations. Our work provides a theoretical basis for studying local disentanglement without assuming a global generative model.

Disentanglement, the separation of factors of variation in data using neural networks, remains a long-standing challenge in machine learning. Prior work has addressed this problem with variational autoencoders and generative adversarial networks that incorporate ideas from variational inference and information-theoretic constraints. In contrast to methods that rely on continuous representations, we propose a design that treats disentangled representations as symbolic structures, motivated by the compositional relationships among the concepts that make up samples from a distribution. However, learning discrete symbolic structures with neural networks while maintaining differentiability is difficult and often requires complex architectures. To address this, we introduce an unsupervised learning algorithm that uses holographic reduced representations (HRR) for neural disentanglement. We show that the HRR unbinding operation provides an inductive bias for separating factors and yields competitive results against baselines, as measured by latent traversals and disentanglement metrics. We complement these empirical findings with an information-theoretic analysis of the HRR unbinding channel. We prove that unbinding induces approximately independent symbol-value pairs and derive a per-slot capacity bound that quantifies how many distinct symbolic concepts can be reliably encoded, giving a quantitative account of the inductive bias toward disentanglement. The resulting representations differ from standard autoencoder-based models, in that their latent units are vectors that are summed together, rather than scalar dimensions of a low-dimensional latent vector. We show that this HRR representation is more robust to noise than other disentangled representations and maintains reconstruction quality across a range of SNRs.