Complete or Sparse: A Tale of Two Identifiabilities

Representation learning models exhibit a surprising stability in their internal representations. Whereas most prior work treats this stability as a single property, we formalize it as two distinct concepts: statistical identifiability (consistency of representations across runs) and structural identifiability (alignment of representations with some unobserved ground truth). Recognizing that perfect pointwise identifiability is generally unrealistic for modern representation learning models, we propose new model-agnostic definitions of statistical and structural near-identifiability of representations up to some error tolerance $ε$. Leveraging these definitions, we prove a statistical $ε$-near-identifiability result for the representations of models with nonlinear decoders, generalizing existing identifiability theory beyond last-layer representations in e.g. generative pre-trained transformers (GPTs) to near-identifiability of the intermediate representations of a broad class of models including (masked) autoencoders (MAEs) and supervised learners. Although these weaker assumptions confer weaker identifiability, we show that independent components analysis (ICA) can resolve much of the remaining linear ambiguity for this class of models, and validate and measure our near-identifiability claims empirically. With additional assumptions on the data-generating process, statistical identifiability extends to structural identifiability, yielding a simple and practical recipe for disentanglement: ICA post-processing of latent representations. On synthetic benchmarks, this approach achieves state-of-the-art disentanglement using a vanilla autoencoder. With a foundation model-scale MAE for cell microscopy, it disentangles biological variation from technical batch effects, substantially improving downstream generalization.

Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering latent structure in data by separating mixed signals into their independent sources. While substantial progress has been made in establishing asymptotic identifiability guarantees for nonlinear ICA, the finite-sample statistical properties of learning algorithms remain poorly understood. This gap poses significant challenges for practitioners who must determine appropriate sample sizes for reliable source recovery. This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders, providing the first complete characterization with matching upper and lower bounds. Our theoretical development introduces three key technical contributions. First, we establish a direct relationship between excess risk and identification error that bypasses parameter-space arguments, thereby avoiding the rate degradation that would otherwise yield suboptimal scaling. Second, we prove matching information-theoretic lower bounds that confirm the optimality of our sample complexity results. Third, we extend our analysis to practical SGD optimization, showing that the same sample efficiency can be achieved with finite-iteration gradient descent under standard landscape assumptions. We validate our theoretical predictions through carefully designed simulation experiments. This gap points toward valuable future research on finite-sample behavior of neural network training and highlights the importance of our validated scaling laws for dimension and diversity.

Methods that recover individual latent variables of the world, from nonlinear ICA to dictionary learning and causal representation learning, anchor the latents to observations through reconstruction, auxiliary supervision, or distributional asymmetries such as non-Gaussianity. Methods without these anchors, including joint-embedding predictive architectures (JEPAs), identify the latent state only up to a linear transformation, so individual latents remain mixed. We close this gap: individual world latents can be provably recovered with no reconstruction, no decoder, and no labels. The key condition is Structural Diversity: different latents leave distinct dependency footprints on observations, just as no two snowflakes are alike. Building on the linear identifiability that LeJEPA provides, we prove that under Structural Diversity, DSReg (Dependency-Sparsity Regularization) recovers individual world latents up to signed permutation, without reconstruction or a decoder. It applies post hoc to any linearly identified representation, reusing trained checkpoints at no loss over joint training, and establishes the first fully identifiable JEPA that recovers every world latent. Moreover, as a condition on dependency footprints, Structural Diversity is strictly weaker than all structural conditions of prior identifiable latent variable models. Across synthetic regimes, world model probes, learned visual encoders, and external renderers, DSReg preserves dense prediction while improving individual-latent recovery and downstream use with scales.