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Anisotropic Representations Improve Planning in JEPA World Models

Mingu Kang, Yoori Oh, Sookyung Kim, Joonseok Lee

arXiv:2609.37441v1cs.RO

TL;DR

Isotropic regularization can produce latent planning costs that misrank outcomes relative to task costs, even with accurate prediction and noncollapsed representations. AnisoWM learns a constrained anisotropic target without changing the prediction objective or Euclidean planner, and improves planning success over LeWM in four visual control environments.

  • Problem

    Isotropic regularization prevents collapse but is not designed to align latent Euclidean distances with task costs, leaving open whether it ranks candidate outcomes in task-cost order.

    Source: §1 · pp. 1–2

  • Method

    AnisoWM jointly learns a constrained diagonal Gaussian target with the encoder and predictor while leaving the prediction objective, predictor architecture, and Euclidean planner unchanged.

    Source: §1 · p. 2

  • Results

    AnisoWM improves planning success over LeWM in all four visual goal-planning environments, and its latent costs show better agreement with task outcomes.

    Source: §1 · p. 2, §6 · p. 10

  • Takeaways & Limitations

    Across the tested environments, changing representation regularization is associated with higher planning success while retaining LeWM’s prediction objective and Euclidean planner.

    Source: §1 · p. 2

    ΛReg uses a diagonal target covariance and does not introduce a full-covariance target parameterization.

    Source: §4.3 · p. 7

Abstract

from arXiv · show

Latent world models learn action-conditioned dynamics in representation space and often score candidate actions by Euclidean distance to a goal representation. Joint training typically regularizes the representation to prevent collapse, but the resulting representation geometry also determines how terminal errors are weighted during planning. We show that accurate prediction and noncollapsed representations do not guarantee a task-aligned latent planning cost: isotropic Gaussian regularization can induce a geometry that ranks feasible outcomes differently from the task cost. To address this mismatch, we introduce AnisoWM with $Λ$Reg, which replaces the fixed isotropic Gaussian target with a learnable diagonal covariance under fixed-trace and anisotropy constraints. The prediction objective, predictor architecture, and Euclidean planner remain unchanged; the target is used only during training. Our analysis characterizes the prediction-driven allocation of target variance, its dependence on the training distribution, and the conditions under which the induced metric reduces planning regret. Across four visual control environments, AnisoWM improves planning success over LeWorldModel in all four. Its latent planning cost also shows better agreement with task outcomes. Project website: https://rkdrn79.github.io/AnisoWM-page/

1 INTRODUCTION

In latent goal planning, the encoder determines how terminal errors are weighted, yet isotropic regularization targets representation quality rather than task-cost alignment. The paper analyzes this mismatch, introduces constrained anisotropic regularization, and reports improved planning success across four visual control environments.

1 INTRODUCTION

  • LeWM rolls out candidate action sequences in latent space and scores them by the distance between predicted terminal and goal representations.The encoder therefore sets the geometry of the planning cost and how terminal errors are weighted.Source: §1 · p. 1
  • SIGReg encourages isotropic Gaussian features to prevent collapse, but its target is motivated by probing-risk criteria rather than planning-cost alignment.The paper frames whether this geometry ranks candidate outcomes in task-cost order as its central question.Source: §1 · pp. 1–2
  • Isotropic SIGReg can yield planning-misaligned geometry despite non-collapse and exact conditional-mean prediction.In the analyzed setting, inverse-state-covariance weighting can reorder feasible outcomes and leave positive regret even as prediction loss vanishes.Source: §1 · p. 2
  • AnisoWM learns a constrained diagonal Gaussian target jointly with the encoder and predictor, while retaining the prediction objective, predictor architecture, and Euclidean planner.The target variance has fixed total variance and bounded anisotropy.Source: §1 · p. 2
  • AnisoWM improves planning success over LeWM in all four environments, and its latent costs agree better with recorded task outcomes.Training learns different target spectra across environments, and allowing more anisotropy does not monotonically improve planning.Source: §1 · p. 2

2 RELATED WORK

Related work spans latent world models, representation regularization, and methods that shape or assess planning geometry. AnisoWM addresses planning geometry through Gaussian representation regularization while retaining the Euclidean planner.

Latent world models

  • Latent world models learn predictive dynamics for planning and control, while LeWM jointly trains an encoder and action-conditioned predictor with SIGReg and uses Euclidean latent distance to plan.Other pipeline modifications include reducing rollout cost and accumulated error or adding multi-horizon prediction and reachability supervision.Source: §2 · p. 2

Representation regularization

  • Representation regularizers constrain feature statistics to prevent collapse or redundancy; LeJEPA derives its isotropic Gaussian target from probing-risk criteria [Balestriero & LeCun, 2025].Other approaches use radial Gaussianization, sparse nonnegative targets, prescribed anisotropic geometry, or temporal centering.Source: §2 · pp. 2–3

Geometry for planning

  • Control-relevant geometry work links latent distances to behavioral equivalence, straightens trajectories, calibrates distances, or learns reachability-aware planning metrics.Decision-Metric Alignment adds latent–outcome ranking diagnostics and action-conditioned objectives; identifiability work studies transformations preserving Euclidean geometry.Source: §2 · p. 3
  • AnisoWM addresses planning geometry through the Gaussian representation regularizer while retaining the Euclidean planner.This contrasts with approaches that replace or augment the terminal planning cost.Source: §2 · p. 3

3 ANALYSIS: ISOTROPIC REGULARIZATION AND PLANNING COST

The analysis shows how isotropic feature regularization induces inverse-covariance weighting, and how that geometry can separate latent planning choices from task-cost choices despite accurate prediction. A nonlinear toy experiment illustrates the same prediction–planning gap.

3.1 THE METRIC INDUCED BY ISOTROPIC COVARIANCE

  • With isotropic encoded covariance, an invertible affine encoder induces M = sΣ−1, so latent Euclidean distance weights state errors by inverse covariance.The latent cost matches a positive-definite task metric for all residuals only when that metric is proportional to Σ−1; lower-training-variance directions receive greater weight.Source: §3.1 · p. 3

3.2 THE METRIC SELECTED BY JOINT PREDICTION–SIGREG TRAINING

  • The analysis formulates joint prediction and SIGReg training with a finite-batch statistic that favors noncollapsed isotropic feature covariance.For sufficiently small prediction-loss weight, the stated proposition gives a global minimum with an invertible encoder and an exact encoded conditional-mean predictor.Source: §3.2 · p. 4
  • As prediction-loss weight η decreases, isotropic regularization selects K → qI_D and hence A⊤A → qΣ−1, despite vanishing prediction loss.The reduction makes the expected regularization term depend only on K, while prediction favors shrinking the encoding along high-noise directions.Source: §3.2 · p. 4

3.3 FINITE-HORIZON PLANNING SEPARATION

  • The finite-horizon construction uses a fully actuated linear Gaussian control family with nonscalar state covariance, bounded actions, and a fixed goal.Its separation result applies for any fixed finite horizon and sufficiently small η under the stated finite-statistic conditions.Source: §3.3 · p. 5
  • Exact conditional-mean rollouts and vanishing prediction and fixed-horizon rollout error can coexist with positive finite-horizon planning regret.For an unreachable goal outside the reachable ball, the task-cost and inverse-covariance latent-cost projections differ when the covariance-weighted goal is not parallel to the goal.Source: §3.3 · p. 5

3.4 ILLUSTRATING THE PREDICTION–PLANNING GAP

  • 30× lower process-noise scale substantially reduced held-out conditional-mean prediction error, while mean normalized physical planning regret remained near 0.16.The observed regret closely matched the 0.1607 inverse-covariance reference, and the learned local pullback metric was close to Σ−1 up to scale.Source: §3.4 · p. 5, Figure 1 · p. 5

4 METHOD

AnisoWM replaces the fixed isotropic target with a constrained learnable variance allocation, while leaving the prediction objective, predictor architecture, and Euclidean planner unchanged. Its analysis connects the learned allocation to the state-space planning metric and identifies conditions for compensating isotropic misalignment.

4 METHOD

  • AnisoWM replaces the fixed isotropic Gaussian target with a learnable diagonal covariance while retaining the prediction objective, predictor architecture, and Euclidean planner.The covariance is used only by the training regularizer and can be discarded afterward.Source: §4 · p. 6, §4.2 · p. 6

4.1 LEARNABLE GAUSSIAN TARGET

  • The target covariance has fixed trace and bounded condition number, while joint prediction–regularization training learns how variance is allocated across latent directions.The trace fixes overall target scale, κ bounds anisotropy, and κ = 1 recovers the isotropic target.Source: §4.1 · p. 6, §4.2 · p. 6

4.2 JOINT TRAINING AND PLANNING

  • Joint training updates the encoder through both losses, the predictor through prediction loss, and the target covariance through regularization; planning remains unchanged from LeWM.The target covariance does not introduce an additional planning-time metric.Source: §4.2 · p. 6

4.3 EFFECT ON PLANNING GEOMETRY

  • With a Euclidean task cost, the learnable target can compensate for isotropic regularization’s inverse-covariance weighting when the required geometry is feasible and selected by predictive training.In the finite-horizon construction, κ = cond(Σ) selects L → Σ, while the isotropic-target planner retains positive limiting regret despite vanishing prediction loss and fixed-horizon encoded rollout error.Source: §4.3 · p. 7

5 EXPERIMENTS

Across four visual goal-planning environments, AnisoWM outperforms LeWM in planning success and improves agreement between latent costs and task outcomes. The experiments also show environment-dependent learned target spectra and non-monotone sensitivity to the anisotropy bound.

5.1 EXPERIMENTAL SETUP

  • The primary comparison uses κ = 2 across TwoRoom, Reacher, PushT, and Cube, with three training seeds and the LeWM baseline trained under identical settings.The evaluation follows LeWM’s datasets, model architecture, and visual goal-planning protocol.Source: §5.1 · p. 8

5.2 RESULTS AND ANALYSIS

  • Planning success is 93% against 87% in TwoRoom, 89% against 86% in Reacher, 97% against 96% in PushT, and 79% against 74% in Cube for AnisoWM versus LeWM.The primary comparison uses κ = 2 in all environments and reports means over three training seeds.Source: §5.2 · p. 8, §5.1 · p. 8
  • AnisoWM improves Jpred action ordering in all four environments; Jenc gains are positive in TwoRoom, Reacher, and Cube, while PushT is nearly unchanged.Jpred is the cost used by CEM and includes predictor rollout; Jenc scores the realized terminal observation without predictor rollout.Source: §5.2 · p. 8
  • AnisoWM’s low-cost neighborhoods more closely follow physical task distance, with Spearman’s ρ rising from 0.15 to 0.91 in Cube and 0.65 to 0.92 in PushT.Figure 3 visualizes the lowest-cost regions around goals for both environments.Source: §5.2 · pp. 8–9, Figure 3 · p. 9
  • At κ = 2, the number of above-mean variance coordinates ranges from 95 in PushT to 124 in Reacher; TwoRoom’s allocation continues evolving after reaching the condition-number bound.The shared anisotropy bound does not determine a single learned spectrum.Source: §5.2 · p. 9, Figure 4 · p. 9
  • Planning success varies non-monotonically with κ, and the highest observed anisotropic setting differs across environments; each κ > 1 point is a single training run.The authors treat this sweep as a sensitivity diagnostic rather than a tuned comparison.Source: Sensitivity to the anisotropy bound · p. 9, Figure 5 · p. 9

6 CONCLUSION

Accurate prediction and noncollapsed representations do not guarantee a task-aligned planning cost. AnisoWM with ΛReg learns a constrained anisotropic target, and the paper reports lower planning regret under analyzed conditions and higher planning success across four visual control environments.

6 CONCLUSION

  • AnisoWM achieves higher planning success than LeWM across four visual control environments, with latent costs showing higher agreement with recorded task outcomes.The analysis characterizes prediction-driven geometry selection and conditions under which it reduces planning regret.Source: §6 · p. 10

B TECHNICAL PRELIMINARIES AND A TWO-DIMENSIONAL CLOSED-FORM SPECIALIZATION

These technical preliminaries relate isotropic latent geometry to inverse state covariance, characterize the finite-batch SIGReg covariance minimum, and give a two-dimensional specialization for analyzing planning regret.

B.1 LATENT METRIC AND ISOTROPIC COVARIANCE

  • For affine encoders, latent Euclidean cost matches a task quadratic cost for all residuals only when its metric is proportional to the inverse state covariance; nonproportional metrics can reverse rankings.A Gaussian marginal constrains a nonlinear encoder's local metric less strongly.Source: §B.1 · p. 15

B.2 EXPECTED FINITE-BATCH SIGREG

  • The expected finite-batch SIGReg statistic has a unique covariance minimum at qI, where q lies between 0 and 1, and its Hessian is positive definite there.The contraction from I to qI reflects finite-batch bias in the squared empirical characteristic-function discrepancy.Source: §B.2 · pp. 16–17
  • The finite-batch minimum condition depends on batch size and quadrature; the sufficient condition includes N = 128, while differing time-slice covariances yield an average of their expected statistics.The stated expectation assumes independent examples within each slice; within-slice dependence requires a different calculation.Source: §B.2 · pp. 16–17

B.3 TWO-DIMENSIONAL CLOSED-FORM SPECIALIZATION

  • In the two-dimensional specialization, isotropic-target training has an invertible minimizing encoder and exact encoded conditional-mean prediction, yet the latent cost can prefer a non-task-optimal action.For sufficiently small positive η, the latent cost strictly prefers uB to 0, while the task cost strictly prefers 0 to uB; the lower-variance coordinate receives greater inverse-covariance weight.Source: §B.3 · p. 18

B.4 TRANSITION CONSTRUCTION AND PREDICTION REDUCTION FOR THE SPECIALIZATION

  • The specialization constructs a covariance-preserving transition whose state and next-state Gaussian marginals match, and whose stationary trajectories provide independent examples within each time slice.Averaged prediction loss and per-slice SIGReg statistics match those of the reset construction for fixed model parameters.Source: §B.4 · p. 19
  • For invertible encoders, the linear encoded conditional-mean predictor attains the prediction bound, and equality uniquely determines that predictor.This reduction lets the specialization compare encoder-induced metrics while retaining exact conditional-mean prediction.Source: §B.4 · p. 19

B.6 GENERAL RESIDUALS AND A FIXED EVALUATION STATE

  • With interior optima and unequal residual coordinates, the learned-target limiting regret is strictly below the baseline exactly when 1 < κ < c2.A uniform positive improvement margin over a residual set additionally requires a positive lower bound on |d1 − d2|.Source: §B.6 · pp. 20–21

B.5 PROOF OF THEOREM 2

The proof establishes a global noncollapsed optimum with exact encoded conditional-mean prediction for small positive η, then derives its latent action and the resulting ranking reversal in the specialization.

B.5 PROOF OF THEOREM 2

  • When ρ1 = ρ2, orthogonal invariance and uniqueness force the reduced minimizer to be scalar, so βη = 0.Under ρ1 < ρ2, the first-order expansion instead gives βη > 0 and c1ρ1 − c2ρ2 < 0.Source: §B.5 · pp. 19–20
  • For sufficiently small positive η, every global minimizer is noncollapsed, uses an invertible encoder, and recovers the exact encoded conditional mean.The reduced covariance minimizer converges to qI, and its diagonal entries remain greater than q/2.Source: §B.5 · pp. 19–20
  • When ρ1 < ρ2, the learned metric's latent minimizer lies in (0, δ) for small η, despite the task-optimal action being 0.The limiting regret of this action approaches 2δ2, while the attained prediction loss tends to zero.Source: §B.5 · p. 20
  • The latent and physical task costs strictly reverse the ordering of the learned action and 0, and this reversal extends to feasible neighborhoods of both actions.Finite-budget CEM performance also depends on candidate sampling and the search procedure.Source: §B.5 · p. 20

C CLOSED-FORM ANALYSIS OF THE LEARNABLE GAUSSIAN TARGET

The two-dimensional analysis makes explicit how a constrained learned target changes the selected planning geometry and regret. It also characterizes a target-coupled rescaling path and spectral bounds for admissible targets.

C CLOSED-FORM ANALYSIS OF THE LEARNABLE GAUSSIAN TARGET

  • Predictive training favors variance reallocation that counteracts the inverse-covariance weighting of the isotropic solution when ρ1 < ρ2.At κ = c, the limiting state-space metric is qI in this specialization, and excessive anisotropy eventually increases regret.Source: C CLOSED-FORM ANALYSIS OF THE… · pp. 21–22
  • For the two-dimensional specialization with ρ1 < ρ2, the learned-target limiting regret is strictly below the isotropic baseline exactly when 1 < κ < c2, and equals it at κ = c2.The limiting regret is zero at κ = c, but becomes higher than the isotropic baseline for κ > c2.Source: C CLOSED-FORM ANALYSIS OF THE… · p. 21

C.2 A TARGET-COUPLED RESCALING PATH

  • The rescaling path preserves standardized features seen by ΛReg while changing encoder geometry, so its changing latent cost reflects the encoder's weighting of process noise.For a fixed anisotropy bound κ, the path belongs to the admissible target set only for 0 ≤ h ≤ (κ − 1)/(κ + 1).Source: §C.2 · pp. 23–24
  • Along the target-coupled rescaling path, regret decreases to zero at h = uB, then rises; sufficiently large h makes it exceed its baseline value.At every h > 0, the expected original isotropic-target regularizer increases by more than the weighted decrease in expected prediction loss.Source: §C.2 · p. 24

C.3 SPECTRAL BOUNDS FOR ADMISSIBLE TARGETS

  • A fixed condition-number bound constrains target effective dimension; at κ = 2, the lower bound is 8D/9.This bound concerns target covariance variance concentration; the trained representation covariance also depends on the encoder.Source: §C.3 · p. 25
  • Every admissible diagonal target can be covered by the logit parameterization, and its eigenvalues and effective dimension obey bounds implied by the trace and condition-number constraints.In two dimensions, trace two and the variance floor vi ≥ 2/(κ + 1) describe the same feasible spectra as the condition-number constraint.Source: §C.3 · p. 25

C.1 PROOF OF THEOREM 3

In the two-dimensional analysis, the learned target changes the limiting planning regret relative to an isotropic baseline according to its anisotropy. The comparison depends on whether κ lies below, at, or above c².

C.1 PROOF OF THEOREM 3

  • The proof localizes every minimizing target and feature covariance around their limiting optima, with the learned covariance converging to qL∗.The positive-definite reduced minimizers are realizable by an encoder, predictor, and diagonal target.Source: §C.1 · p. 23
  • For sufficiently small noise, the unconstrained minimizing action is feasible and unique because the limiting action lies strictly inside the action bounds and the quadratic cost is strictly convex.The limiting regret follows by evaluating the task cost at that action.Source: §C.1 · p. 23
  • Limiting regret is lower than the baseline for 1 < κ < c², equal at κ = c², and higher for κ > c²; it vanishes at κ = c.When the limiting improvement is positive, the regret improvement also holds for sufficiently small positive noise.Source: §C.1 · p. 23

C.4 SCOPE OF THE ANALYSIS

The analysis makes the regret comparison conditional on the control model and task metric, and characterizes how the training noise structure selects target spectra. In arbitrary dimension, sufficiently small-noise optima have a constrained two-value spectrum whose multiplicity and orientation depend on that structure.

C.4 SCOPE OF THE ANALYSIS

  • With ρ1 < ρ2, every κ > 1 gives strictly higher limiting regret than the baseline, whereas for ρ1 = ρ2 all target orientations tie in the leading trace term.When ρ1 > ρ2, the minimizing orientation reverses; in two dimensions the target spectrum is determined by κ up to permutation, unlike the higher-dimensional multiplicity selection.Source: §C.4 · p. 26
  • For the Q-weighted task cost, choosing κ = κQ gives zero limiting regret when κQ > 1 under ρ1 < ρ2.The calibration uses the state covariance and task metric; Q = I recovers κQ = c.Source: §C.4 · p. 26
  • The asymptotic comparisons hold for fixed, strictly positive regularization weights as η ↓ 0; weights that scale with η define a different regime.Source: §C.4 · p. 26

D PREDICTION-DRIVEN TARGET SPECTRUM SELECTION

D.1 SPECTRUM SELECTION IN ARBITRARY DIMENSION

  • For sufficiently small η, every global optimum has a two-value target spectrum with condition number κ; its larger-variance directions align with the lowest-noise eigendirections.The multiplicity minimizes the trace criterion, and a unique minimizing index with an eigengap yields a unique limiting target geometry.Source: §D.1 · pp. 27–28
  • When trace criteria tie, the next-order criterion selects among tied spectra; for D = 3, κ = 2, and R = diag(1, 2, 4), it selects the m = 1 branch for sufficiently small positive noise.Source: §D.1 · p. 28

D.2 TWO TRAINING DISTRIBUTIONS WITH DIFFERENT SELECTED SPECTRA

Two three-dimensional training models with identically ordered noise spectra select different target allocations, yet each yields zero limiting task regret. The accompanying analysis connects these examples to the geometry induced by isotropic joint training and its regularizer assumptions.

D.2 TWO TRAINING DISTRIBUTIONS WITH DIFFERENT SELECTED SPECTRA

  • With κ = 2, the two training distributions select target allocations with one versus two larger-variance directions despite identical noise-eigenvalue ordering, and both have zero limiting task regret.The learned optima are noncollapsed for sufficiently small positive noise.Source: §D.2 · pp. 29–30
  • The examples illustrate distribution-dependent spectrum selection: relative noise magnitudes, not just their ordering, determine how many target directions receive the larger variance.Source: §D.2 · pp. 28–30

E GENERAL-DIMENSIONAL GEOMETRY, FINITE-HORIZON PLANNING, AND NONLINEAR STABILITY

  • The appendix establishes general-dimensional joint-optimum results and gives a finite-horizon construction, with learned-target and nonlinear extensions also recorded.Source: E GENERAL-DIMENSIONAL GEOMETRY,… · p. 30

E.1 STANDING REGULARIZER ASSUMPTIONS

  • The finite-batch regularizer used in the analysis has a unique isotropic minimum qI and a positive-definite Hessian on symmetric matrices.Source: §E.1 · p. 30

E.3 PROOF OF PROPOSITION 2

The appendices establish conditions under which learned target geometry supports task-aligned planning and its regret advantage over isotropic training persists, with bounds on stability and perturbations under stated assumptions. They distinguish linear-model results from nonlinear encoder rollouts and outline the experimental protocol and additional diagnostics.

E.3 PROOF OF PROPOSITION 2

  • Proposition 2’s proof establishes a global minimum and shows that every full global minimizer has an invertible encoder and exact encoded conditional-mean predictor; its limiting metric depends on a feasible, predictively selected geometry.When the trace minimizer is unique, the learned metric converges to the limit in Eq. (12); the task-alignment condition requires the selected geometry to be proportional to the task metric.Source: §E.3 · p. 32

E.4 EXACT ROLLOUTS AND FINITE-HORIZON REGRET

  • At global optima, exact latent rollouts predict encoded conditional means, and finite-horizon planning regret converges uniformly to the mismatch between the learned-metric and task-metric terminal minimizers.The limiting regret is positive exactly when the two terminal minimizers differ; the common task-noise term cancels from open-loop regret.Source: §E.4 · p. 33

E.5 SEPARATION IN EVERY DIMENSION AND AT EVERY FINITE HORIZON

  • Across every dimension and fixed finite horizon in the construction, learned-target regret is strictly lower than isotropic regret at sufficiently small positive noise, although both methods have vanishing training losses and fixed-horizon encoded rollout MSE.The comparison is regret relative to the best feasible sequence, not zero terminal error; the goal lies outside the limiting reachable ball, and the noise threshold may depend on the construction parameters.Source: §E.5 · pp. 34–35

E.6 STABILITY OF THE PLANNING SEPARATION

  • The regret separation is stable under cost perturbations and approximate planning: if ϵJ < ∆/16 and 2ϵT + ζT < q∆/8, the first regret exceeds ∆/2 while the second is less than ∆/4.The lemma also requires 2ϵB + ζB < γ for its stated separation condition.Source: §E.6 · p. 35

E.7 SMOOTH NONLINEAR PERTURBATIONS

  • In a sufficiently small smooth nonlinear neighborhood of the constructed systems, the regret separation persists even with exact one-step conditional-mean prediction and exact sequence optimization.The result also gives globally invertible encoders with uniformly nonsingular Jacobians and uniformity over the reference global training optima.Source: §E.7 · p. 36

4. Applying the limiting regret margin. Define on the same compact action set

  • The nonlinear separation holds uniformly over allowed training optima and planning choices, including nonunique or approximate minimizers, when the perturbations and optimization error satisfy the stated smallness thresholds.The proof applies the quantitative regret-margin lemma after making the latent and physical cost perturbations sufficiently small.Source: §4 · p. 37
  • The corollary proves stability of planning behavior near jointly selected linear models, not selection of nearby encoders by unrestricted nonlinear joint optimization.For nonlinear encoders, an exact one-step encoded conditional mean need not compose into an exact terminal conditional mean; an explicit rollout bound handles this distinction.Source: §4 · p. 37
  • Global perturbation bounds accommodate unbounded Gaussian innovations, since compact action constraints alone do not bound noisy state trajectories.Source: §4 · p. 37

F EXPERIMENTAL DETAILS

  • The appendix reports the evaluation protocol, per-seed planning results, and the latent-cost ordering diagnostic used in Table 1.It also examines ΛReg target allocation, sensitivity to the anisotropy bound, and additional representation and prediction diagnostics.Source: F EXPERIMENTAL DETAILS · p. 38

F.1 EXPERIMENTAL PROTOCOL AND PER-SEED RESULTS

The appendix reports latent-cost ordering diagnostics, environment-dependent target allocation, and anisotropy-bound sensitivity. It also examines realized feature geometry and prediction-loss relationships, while treating rollout pairs as illustrations rather than aggregate performance evidence.

F.1 EXPERIMENTAL PROTOCOL AND PER-SEED RESULTS

  • Across TwoRoom, Reacher, PushT, and Cube, AnisoWM uses LeWM’s datasets and preprocessing protocol, 192-dimensional representations, and, unless otherwise stated, κ = 2 and λ = 0.09; LeWM uses κ = 1 with the same architecture and optimizer settings.Source: §F.1 · p. 38

F.2 LATENT-COST ORDERING DIAGNOSTIC

  • AnisoWM ranks outcomes better under both Jenc and Jpred in TwoRoom and Reacher, while PushT improves only under Jpred and is nearly unchanged under Jenc.Figure 7 compares the models' costs on the same recorded sequences.Source: §F.2 · p. 41, Figure 7 · p. 41

F.3 ADDITIONAL LOCAL COST GEOMETRY

  • Across four additional Cube and PushT examples, AnisoWM has higher Spearman rank correlation between latent planning cost and task cost than LeWM.Figure 8 marks task-cost neighborhoods in physical space and latent-cost regions for the two models.Source: §F.3 · p. 41, Figure 8 · p. 42

F.4 LEARNED TARGET ALLOCATION

  • At κ = 2, above-mean target-variance coordinates number 102, 124, 95, and 97 in TwoRoom, Reacher, PushT, and Cube; TwoRoom's spectrum keeps evolving after reaching its bound.In TwoRoom sweeps, bounds κ ≤ 8 are reached, but the κ = 16 run ends at condition number 11.3.Source: §F.4 · pp. 41–42

F.5 SENSITIVITY TO THE ANISOTROPY BOUND

  • Preferred anisotropy bounds differ by environment: TwoRoom improves across the displayed range, while other environments attain their best anisotropic result at smaller bounds.These sensitivity sweeps use single training runs and are not replacements for the three-seed primary comparison.Source: §F.5 · p. 43

F.6 REALIZED REPRESENTATION GEOMETRY

  • In TwoRoom, normalized feature-covariance eigenvalue 100 rises from 0.009 at κ = 1.25 to 0.054 at κ = 16; entropy effective rank ranges from 74 to 111 of 192 dimensions.The measured feature covariance is distinct from the learned target covariance.Source: §F.6 · pp. 43–44, Figure 10 · p. 44

F.7 ADDITIONAL PREDICTION DIAGNOSTICS

  • The run with the lowest latent MSE is not consistently the run with the highest planning success, and each run's MSE is measured in its own representation coordinates.The comparison is a diagnostic, not a common physical prediction-error measure.Source: §F.7 · p. 44, Figure 11 · p. 45

F.8 QUALITATIVE ROLLOUTS

  • Figures 12 and 13 show recorded rollout pairs in all four environments under the same evaluation planner; these examples illustrate behavior rather than aggregate performance.The pairs compare AnisoWM at κ = 2 with LeWM under matched seeds, with red outlines marking recorded goal attainment.Source: §F.8 · p. 44, Figure 12 · p. 46, Figure 13 · p. 47

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