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Block Disentanglement in CRL: Bridging Identifiability and Visual State Estimation

Emre Acartürk, Pranamya Kulkarni, Puranjay Datta, Karthikeyan Shanmugam, Burak Varıcı, Ali Tajer

arXiv:2610.06809v1cs.LGstat.ML

TL;DR

Recovering causal variables from visual observations is difficult when labels are unavailable and existing identifiability guarantees depend on strong assumptions. This paper establishes block-identifiability conditions under weaker intervention assumptions and evaluates the approach in embodied visual state estimation. It reports that the block-based principle remains informative under further assumption violations in a simulated robot setting.

  • Problem

    Recovering hidden physical variables and causal mechanisms from high-dimensional observations is important, but existing CRL guarantees often rely on assumptions that may not hold together in practical visual settings.

    Source: §1 · pp. 1–2

  • Method

    The paper replaces individual-variable identifiability with block disentanglement under weaker intervention conditions, then applies a score-based objective to embodied visual data.

    Source: §1 · pp. 2–3, §4.2 · p. 8

  • Results

    In simulated robot pose estimation, the method retains significant information about the active block despite temporal dependence, imperfect intervention effects, and uncertainty in the latent representation.

    Source: §1 · p. 3, §6 · p. 12

  • Takeaways & Limitations

    Block disentanglement provides an identifiable target under weaker intervention assumptions, and the resulting principle remains informative in the paper’s embodied visual setting despite further departures from theory.

    Source: §1 · p. 3, §6 · p. 12

    The latent causal model assumes an injective smooth observation map, full-support latent distribution, and continuously differentiable causal mechanisms.

    Source: §2 · p. 3

Abstract

from arXiv · show

Causal representation learning (CRL) is the process of recovering causally-related latent variables from high-dimensional observations. As a label-free inference method, CRL is particularly attractive for applications where data labels are unavailable or impractical to obtain. While there has been significant progress in understanding the identifiability guarantees of CRL, such guarantees often hold under highly stylized assumptions, which temper the direct application to real-world problems. This paper has a two-fold objective for interventional CRL. First, it establishes identifiability guarantees for substantially weaker interventional assumptions, resulting in block disentanglement of the causal variables, where the block structure depends on the realistically available intervention mechanisms. Secondly, the block disentanglement framework is used for embodied visual state estimation, in which the objective is to recover the latent physical variables of a robotic system directly from visual data (images and videos) without labeled data. These two components are critically complementary. The block disentanglement theory delineates identifiability guarantees under weakened assumptions, and the application demonstrates that the resulting objective remains effective in a controlled embodied setting despite further assumption violations, providing a theory-to-practice bridge needed to translate the promise of label-free CRL into practical problems.

1 Introduction

CRL aims to recover hidden causal variables and mechanisms from high-dimensional observations, including unlabeled video, but existing identifiability guarantees rely on assumptions that may not hold together in practical settings. The paper weakens intervention assumptions to guarantee block disentanglement, then examines whether the principle remains informative for embodied visual state estimation under further assumption violations.

Motivation

  • Recovering hidden physical variables and causal mechanisms from sensory observations matters for reasoning about hypothetical stimuli and transferring models across environments with distributional shifts.The paper motivates visual state estimation from unlabeled video because frame-by-frame annotation is costly.Source: §1 · pp. 1–2
  • Existing CRL guarantees often jointly require counterfactual observations, atomic hard interventions, independent samples, exhaustive intervention diversity, and known latent dimensionality, conditions that may not hold in evolving systems.Sequential data and partial or imperfect interventions with unknown targets are examples of departures.Source: §1 · p. 2

Theoretical bridge

  • The theory establishes conditions for identifying an intervened block even when its individual causal variables cannot be disentangled separately.This relaxes the identifiability target from individual variables to a block.Source: §1 · pp. 2–3

Empirical bridge

  • In a simulated robot pose-estimation task, the method retains significant information about the active variable block despite temporal dependence, imperfect intervention effects, and latent-representation uncertainty.The experiments investigate embodied visual data under these assumption violations.Source: §1 · p. 3

Overall contribution

  • The paper reports that block disentanglement remains effective beyond stringent stylized assumptions, with theory and experiments addressing progressively weaker conditions.The theoretical result weakens the identifiability assumptions; the empirical setting departs further from those requirements.Source: §1 · p. 3

2 Preliminaries

The preliminaries set up a latent causal model and distinguish elementwise recovery from the paper’s block-level identifiability target. They frame the contribution as relaxing intervention assumptions in two stages, with empirical work going beyond the theoretical conditions.

Latent causal model

  • The model maps latent variables through an injective, smooth diffeomorphism to observations, while the latent distribution factorizes according to a DAG and its mechanisms are continuously differentiable.The latent distribution is assumed to have full support on R^n.Source: §2 · p. 3

CRL block disentanglement

  • Elementwise disentanglement recovers variables individually, whereas block disentanglement targets an invertible representation of a true block without leakage from other variables or noise.The paper focuses its theoretical analysis on the block target under weaker data-availability conditions.Source: §2 · p. 4

Weakened assumptions

  • The paper organizes prior intervention assumptions into six categories: atomic targets, hard interventions, pairing, intervention discrepancy, sample independence, and known latent dimension.These categories provide a common reference for comparing assumptions across studies.Source: §2 · pp. 4–5
  • Prior general nonlinear results establish elementwise disentanglement under assumptions C1–C6, while this paper replaces C1–C4 with weaker conditions and empirically also relaxes C5–C6.The theoretical stage guarantees block disentanglement; the empirical stage tests visual-data settings departing further from the theory.Source: §2 · p. 5

3 Theoretical Bridge: Weakening Intervention Diversity

The theoretical framework replaces intervention-by-intervention restrictions with aggregate conditions on an active block. Under these conditions, and assumptions C5–C6, a suitably supported encoder representation block-disentangles the intervened latent block.

Score-based framework

  • The framework uses latent and observed score functions and their intervention-induced differences to formalize sufficient conditions for block disentanglement.The observed score differences are related to latent score differences through the observation-map Jacobian.Source: §3 · pp. 5–6

Intervention conditions

  • Ancestral closure confines intervention-induced score differences to the active block, while collective rank diversity requires those differences jointly to span that block.Together these aggregate conditions replace canonical restrictions on individual interventions.Source: §3 · p. 6
  • Collective rank diversity permits unpaired, multi-node, and soft interventions when their score-difference directions span the active subspace, unlike canonical full-coverage structures.The condition is satisfied by at least |I| environments with score-difference directions spanning the active subspace almost everywhere.Source: §3 · p. 6

Block identifiability

  • Under ancestral closure, collective rank diversity, C5–C6, perfect reconstruction, and the stated row-support condition, the encoder representation block-disentangles the active latent block.The theorem requires the estimated matrix to be supported on a row set whose size equals the active block size.Source: §3 · p. 6

Comparison to prior work

  • Unlike the invariance-based framework of Yao et al. (2025), this paper targets the intervention-affected, or varying, block.The cited invariance-based approach identifies latent blocks that remain invariant.Source: §3 · p. 7

4 Empirical Bridge: Block Disentanglement to Embodied Video

The empirical bridge compares a benchmark satisfying assumptions C1–C6 with an embodied visual setting that additionally relaxes C5–C6. The section describes the robot data and score-based objectives used to assess visual state estimation and active-block information retention.

4 Empirical Bridge: Block Disentanglement to Embodied Video

  • The assumption-matched benchmark satisfies C1–C6, while the embodied setting additionally departs from C5–C6 to assess the approach under weaker conditions.The comparison benchmarks performance under canonical assumptions and tests the reach of the theory-to-practice bridge.Source: §4 · p. 7

4.1 Setup and Data Generation

  • The study estimates a robot arm’s joint configuration from rendered images, varying joint angles while fixing other factors and using two camera viewpoints to support observability.Images are generated with Panda-Gym; the two viewpoints support the injective-rendering assumption.Source: §4.1 · p. 7
  • The embodied setting uses continuous controller-driven videos, with environments defined by goal-sampling distributions that prioritize a target joint’s displacement.It introduces within-video temporal dependence, imperfect coupled intervention effects, and a mismatch between learned bottleneck and true latent dimensions.Source: §4.1 · p. 8

4.2 Methodology

  • In the assumption-matched benchmark, a score-based regularizer and reconstruction loss use paired intervention score differences to guarantee elementwise disentanglement under the stated assumptions.The paired score difference is one-sparse on the true latent domain.Source: §4.2 · pp. 8–9
  • The embodied objective confines score differences to a fixed active block and penalizes components outside the estimated block.The method estimates score differences directly from trajectories and uses the change-of-variables formula.Source: §4.2 · p. 9

5 Empirical Results and Analysis

The experiments evaluate label-free visual state estimation in an assumption-matched benchmark and an embodied setting that violates theoretical assumptions. Results show that score regularization preserves state information in the active block under those violations, while active predictability remains stable across bottleneck dimensions.

Experiment pipeline

  • A binary classifier estimates log-density ratios between selected distributions, and its logit gradient estimates feature-space score differences.AE2 uses canonical score sparsity in the assumption-matched benchmark and block sparsity in the embodied setting.Source: §5 · p. 9
  • Figure 1 shows each camera image compressed to 8×8×1 features, fused into an 8×8×2 representation, and mapped by AE2 to a db-dimensional latent estimate.The implementation uses an initial autoencoder for dimensionality reduction, followed by a learned representation stage.Source: §5 · p. 9, Figure 1 · p. 10

Evaluation metrics

  • Assumption-matched recovery is evaluated with affine-calibrated held-out MSE, while embodied active-block recovery uses nonlinear-kernel-ridge explained variance and compares information across pipeline stages.The comparisons localize information loss to feature extraction, representation learning, or active-block construction.Source: §5 · pp. 10–11

Assumption-matched setting results

  • The label-free CRL approach achieves performance comparable to RoboPEPP trained with labels for 5% of the dataset.That training subset is approximately 6.5K labeled poses, or 13K labeled images from two camera orientations.Source: §5 · p. 11, Table 1 · p. 11

Embodied setting results

  • At db=32, active predictability is 0.61 with λ=50 versus 0.23 with λ=0, against all-latent predictability of 0.67.The embodied experiments activate violations V1–V3; the reported comparison shows the block regularizer retains much of the state information in the full latent representation.Source: §5 · p. 11, Table 2 · p. 12

Sensitivity to assumption violations

  • Across db ∈{8, 12, 16, 32}, active predictability remains essentially unchanged even as the full representation learns additional reconstruction-helpful features.The active/nuisance contrast persists under substantial overparameterization.Source: §5 · p. 12, Table 2 · p. 12

6 Concluding Remarks

The paper proposes a two-step path from weaker-assumption identifiability theory to label-free visual state estimation. It establishes block-disentanglement guarantees and reports that the principle remains informative in embodied settings with further assumption violations.

Theoretical contribution

  • Weaker interventional assumptions yield block-disentanglement guarantees: an active group of causal variables can remain identifiable even when individual variables cannot be recovered separately.The block structure reflects what remains identifiable under the relaxed assumptions.Source: §6 · p. 12

Empirical contribution

  • In embodied visual settings, the block-based objective remains informative about relevant physical variables despite temporal dependence, imperfect interventions, and overcomplete representations.The conclusion frames the theory and experiments as complementary steps toward label-free visual state estimation.Source: §6 · p. 12

Next step

  • Extending identifiability theory to temporal dependence, approximate intervention effects, and unknown or overcomplete latent dimensionality remains a key next step.Source: §6 · p. 12

AI Use Statement

The authors report using generative AI tools in implementation and research-support tasks, while excluding them from developing theoretical models and mathematical claims.

Use in research

  • Generative AI tools were used to implement methods and provide feedback on research methodology or experiments.Source: AI Use Statement · p. 13

Uses and exclusions

  • The authors report using AI for code, parts of the draft, literature analysis, brainstorming, and readability edits, but not for theoretical models, mathematical claims, or proofs.Source: AI Use Statement · p. 13

Review and responsibility

  • The authors reviewed AI-assisted work, tested LLM-generated code, and accept responsibility for the final content.Source: AI Use Statement · p. 13

A Related Work and Limitations

The related work contrasts intervention-defined block recovery with elementwise CRL and supervised or geometry-based pose estimation. The discussion also identifies boundaries in the theory and the controlled scope of the empirical evidence.

Related work

  • Prior CRL identifiability work spans linear and nonlinear observation models and hard, soft, known, or unknown interventions, often targeting elementwise recovery.Block objectives include recovering groups up to invertible within-group transformations, as in the paired-observation content-block setting.Source: A Related Work and Limitations · pp. 17–18
  • Collective score diversity identifies the intervention-affected block even when its individual coordinates cannot be separated, replacing intervention-by-intervention diversity requirements with collective conditions.Earlier block guarantees use settings such as known target sets with diverse perfect interventions or particular soft-intervention structures.Source: A Related Work and Limitations · p. 18
  • Visual pose methods such as DREAM and RoboPose rely on pose or joint-state supervision, whereas this work uses intervention-environment variation without per-frame state labels.RoboPEPP also uses self-supervised visual pretraining, but its joint-angle regression is supervised.Source: A Related Work and Limitations · p. 18
  • Geometry- and marker-based pose pipelines rely on known geometry, calibration, or instrumentation; the paper positions robot pose as a controlled testbed, not a replacement for them.The stated aim is to study whether identifiable-CRL principles carry toward state estimation under progressively weaker assumptions.Source: A Related Work and Limitations · p. 18

Limitations

  • The empirical analysis is limited to controlled simulation, and the authors call for real-world data while stating that the transition need not be abrupt.The controlled setting is a deliberate choice to maximize experimental controllability.Source: A Related Work and Limitations · pp. 18–19
  • Video temporal dependence remains theoretically unaddressed until both score estimation and disentanglement are robust to non-i.i.d. data; off-target effects need an approximate recovery bound.The proposed bound would relate block-recovery degradation to the magnitude of off-block score deviations.Source: A Related Work and Limitations · p. 19
  • Consistent joint occlusion from an unfavorable camera angle impedes recovery and requires a multi-camera setup; empirically, the method is robust to an overcomplete representation.The overcomplete-bottleneck observation concerns empirical robustness, while other CRL approaches can also identify the latent dimension.Source: A Related Work and Limitations · p. 19

B Proofs

The appendix proves a block-identifiability result and details embodied experiments using temporally dependent visual data with imperfect intervention effects. Its sensitivity analyses examine how bottleneck size and score regularization affect active-block predictability.

B Proofs

  • Under the theorem’s conditions, perfect reconstruction and the specified support of the estimated matrix imply that the estimate block-disentangles the true active block.The proof establishes that the block map is a diffeomorphism.Source: B Proofs · pp. 19–20

C Embodied Regime

  • The embodied experiments use six environments and designated active coordinates; ground-truth joint angles are reserved for evaluation rather than representation training.When db > 6, the additional learned coordinates are treated as nuisance coordinates.Source: C Embodied Regime · p. 20

C.1 Data Generation

  • Each environment emphasizes displacement of one joint, but continuous closed-loop trajectories produce temporally dependent frames and generally move other joints too.Thus, the environment label indicates the emphasized joint, not an intervention on only one latent coordinate.Source: §C.1 · p. 21

C.2 Score Difference Estimation

  • The log-density-ratio estimator uses non-overlapping temporal halves as an empirical proxy for the theoretical interventional score difference, not a direct estimate of it.The halves provide a coarser, more separable contrast than heavily overlapping portions of a trajectory.Source: §C.2 · p. 22

C.3 Training and Evaluation Details

  • Evaluation reports feature, all-latent, and active predictability using nonlinear kernel ridge regression on held-out labeled states, which are not used for representation learning.The bottleneck sweep fixes λ = 50; the score-regularization sensitivity experiments vary λ.Source: §C.3 · p. 22

C.4 Sensitivity and Additional Results

  • The sensitivity experiments vary bottleneck dimension and score-regularization weight to show training dynamics underlying the aggregate results in Table 2.The finalized sweeps cover db and λ.Source: §C.4 · pp. 22–23

C.4.1 Bottleneck Dimension

  • As db increases, active predictability remains comparatively stable and close to the information available from the complete learned representation.The text attributes this pattern to concentration of recoverable state information in the designated six-dimensional subspace.Source: Figure 2 · p. 23, §C.4.1 · p. 23

C.4.2 Score-Regularization Weight

  • The reconstruction-only baseline does not consistently concentrate state information in the active block; block regularization closes this gap, and increasing λ beyond a useful range does not necessarily improve active predictability.The reported training dynamics span db = 8, 16, and 32.Source: Figure 3 · p. 23, §C.4.2 · pp. 23, 25, Figure 4 · p. 24

C.4.3 Multi-Dominant-Joint Trajectories

The multi-dominant-joint results are nearly indistinguishable from the single-dominant-target regime, consistent with the block-based objective remaining useful beyond atomic interventions. The appendix also describes the embodied datasets, training, and evaluation results across camera setups and causal joint data.

C.4.3 Multi-Dominant-Joint Trajectories

  • Multi-target results are almost indistinguishable from the single-dominant-target regime, while the score-based block regularizer concentrates recoverable state information and remains relatively insensitive to overparameterization.The experiment increases off-target effects from 0.05 to 0.1, testing interventions distributed across multiple coordinates.Source: §C.4.3 · p. 25

D.1.2 Causal Dataset Generation

  • The inference data comprise observational and interventional distributions for each joint, and the learning algorithm uses images without distribution metadata or statistics.The causal dataset samples root joints and determines the others through linear functions of their ancestors plus noise; interventions directly set a joint to a sampled new value.Source: §D.1 · p. 26, §D.1.2 · p. 28, Table 6 · p. 30

D.1 Data Generation

  • The dataset uses one camera for in-plane joints 2, 4, and 6, then two camera angles to capture all six joints, including out-of-plane motion.The two-camera data points contain two observational images and 24 interventional images; each joint receives two hard interventions.Source: §D.1 · pp. 26–27

D.1.1 Independent Interventional Distributions

  • For two-camera experiments, the independent and occlusion models train on ID data and are tested on a separate OOD set with changed observational parameters.Interventional distributions remain identical between the ID and OOD datasets.Source: §D.1.1 · p. 27, Table 4 · p. 28, Table 5 · p. 29

D.2 Training Details

  • Training comprises Autoencoder-1, per-joint log-ratio density estimators, and per-joint Autoencoder-2; evaluation reports MCC and MSE, with MSE calibrated by elementwise affine regression on 1,000 held-out labeled samples.Scatter plots show the highest-MCC trial out of 15, complementing aggregated statistics in the main results tables.Source: D Assumption-Matched Regime · p. 25, §D.2 · p. 30, §D.3.4 · p. 32

D.3.1 Single-Camera: Independent Joints

  • In the single-camera setup, MCC is consistently at least 0.94 for the three visible in-plane joints, 2, 4, and 6.The paper reports this result as stronger than prior CRL experiments on simpler image datasets (Varıcı et al., 2025, Table 14).Source: §D.3.1 · p. 31

D.3.2 Extending to Multiple Views: Two Cameras with Independent Joints

  • The two-camera independent model disentangles all six joints, though joints 3 and 5 perform worse; MCC for joints 2, 4, and 6 remains robust across camera setups.The authors attribute the weaker joints 3 and 5 results to less precise LDR score estimates, supported by higher classification loss.Source: §D.3.2 · pp. 31–32

D.3.3 Two-Camera with Causal Model over Joints

  • Introducing a linear causal data-generation process generally improves performance, with MSE decreasing across most joints versus the independent two-camera model.The causal dataset uses randomly chosen structure and edge weights, with joints 1, 2, and 4 as roots.Source: §D.3.3 · p. 32

D.4 Supervised Comparison and Robustness

The score-based method shows robustness to unseen occlusions, including inpainting occluded regions in its final reconstruction. The section also compares it with RoboPEPP and clarifies that scatter plots show best trials while tables report means.

Supervised Comparison

  • RoboPEPP predicts six joint angles from concatenated embeddings of two camera views and is trained with ground-truth angle labels.Its first stage self-supervises an I-JEPA backbone; the second trains JointNet with an L2 loss.Source: §D.4 · pp. 32–33

Qualitative and Quantitative Results

  • Figures 10 and 11 show the best trial out of 15 for each joint, whereas the reported tables give mean statistics across runs.The scatter plots pair learned latent variables with their corresponding ground-truth joint angles.Source: Figure 10 · p. 33, Figure 11 · p. 34, Table 9 · p. 35

Supervised Comparison

  • The method achieves performance comparable to RoboPEPP trained with 5% of the dataset’s labels on joints 1, 2, 4, and 6.That 5% corresponds to approximately 6.5K labeled poses or 13K images from two camera orientations.Source: §D.4 · p. 33

Robustness to Occlusion

  • Despite no occlusion training, the method degrades less than RoboPEPP on joints 2 and 4 and has lower overall MSE at both 10% and 100% label-data settings.AE2’s final reconstruction inpaints the occluded region, although AE1’s reconstruction still shows the patch.Source: §D.4 · p. 34, Figure 12 · p. 35

D.5 Ablations

The ablations find that score-based sparsity loss is central to disentanglement, while smaller learning rates improve performance and larger batches converge faster. Qualitative examples show intervention views and reconstruction stages across camera setups.

D.5 Ablations

  • Learning rates of 5e−5 and 8e−5 lead to better performance than the other tested rates.The comparison fixes batch size at 128 and sparsity-loss weight at λ = 3.Source: §D.5 · p. 35, Figure 13 · p. 38
  • Larger batch sizes converge faster, but final asymptotic performances are nearly identical across batch sizes.The comparison uses the previously selected learning rate of 5e−5.Source: §D.5 · p. 35, Figure 14 · p. 39
  • MCC exceeds 0.9 with substantial score-based loss weight but remains below 0.15 with reconstruction loss alone.The ablation varies reconstruction and sparsity weights at a fixed learning rate of 5e−5 and batch size of 128.Source: §D.5 · p. 39, Figure 15 · p. 40
  • Intra-module residual connections improve reconstruction loss and, consequently, final MCC.The connections are applied independently within the encoder and decoder.Source: §D.5 · p. 39, Figure 16 · p. 40
  • MSE remains stable when calibration samples are reduced from 1,000 to 100, while dark lighting most strongly degrades MCC for joints 5 and 6.Other joints retain strong performance with minimal degradation under the tested dark and light conditions.Source: §D.5 · pp. 39, 41, Table 13 · p. 41, Table 14 · p. 41

D.6 Qualitative Examples

  • Intervention images show Joint 4 interventions in the single-camera dataset and Joint 3 interventions from two camera angles in the two-camera dataset.The authors identify the two-camera setup as necessary for Joint 3’s out-of-plane motion.Source: §D.6 · p. 41, Figure 18 · p. 42, Figure 17 · p. 42
  • AE1 reconstruction quality is slightly worse with two-camera data than with single-camera data, and that reduction carries through to AE2 reconstructions.The authors hypothesize that the multi-view dataset’s greater visual variance makes reconstruction more challenging.Source: §D.6 · p. 41

E Architecture Details

The appendix specifies the architectures of AE1, the log-density-ratio network, and AE2, and provides visual comparisons of reconstruction stages for single-camera and two-camera independent models.

E Architecture Details

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