When Does LeJEPA Learn a World Model? — part 9/9

Cleaned-text code points [124914, 129109). Bibliography omitted here; source.md includes it.

Fig. 15 resolves identifiability by joint. OU recovers shoulder and wrist symmetrically, consistent with the isotropy of the Gaussian OU transition (see App. F for why isotropic transitions are necessary for the simultaneous approach). Trajectory recovery is massively asymmetric: the wrist, driven by a nearly-bimodal policy-induced marginal and a faster autocorrelation, is recovered poorly at small δ \delta and improves only once δ \delta is large enough to provide informative temporal variation; the shoulder degrades at large δ \delta due to wrapping beyond ± π \pm\pi . This is the empirical signature predicted by the Sturm–Liouville analysis when isotropy fails.

Fig. 16 connects back to the approximate bound. Left: OU identifiability is robust to λ \lambda at high ρ \rho ; only at ρ = 0.99 \rho=0.99 with the strongest λ \lambda does SIGReg begin to dominate alignment, mirroring the collapse regime of the grid search (Fig. 6 ). Right: the fitted orthogonality error ‖ Q ^ ⊤ ​ Q ^ − I ‖ F / n \|\hat{Q}^{\top}\hat{Q}-I\|_{F}/\sqrt{n} decreases monotonically with ρ \rho , tracking the tightening spectral gap 2 ​ ρ ​ ( 1 − ρ ) 2\rho(1-\rho) that governs Thm. 5.3 .

### H.12 Planning Experiment

The planning experiment (Sec. 6.4 , Fig. 5 ) evaluates straight-line latent interpolation across all trained Reacher encoders (App. H.11 ). The qualitative panels (Fig. 5 left) compare the best Gaussian encoder (OU condition at ρ = 0.99 \rho=0.99 ) against the best Trajectory encoder (stride δ = 8 \delta=8 ); the quantitative panels (middle, right) aggregate over the full set.

##### Setup.

We construct a retrieval library of 10,000 10{,}000 held-out evaluation frames paired with their joint-angle labels. Given a start frame x s x_{s} and a goal frame x g x_{g} , we encode both to latents y s = f ⁡ ( x s ) y_{s}=f(x_{s}) and y g = f ⁡ ( x g ) y_{g}=f(x_{g}) and form the straight-line latent interpolant y t = ( 1 − t ) ​ y s + t ​ y g y_{t}=(1-t)\,y_{s}+t\,y_{g} for t ∈ { 0 , 1 T , … , 1 } t\in\{0,\tfrac{1}{T},\ldots,1\} with T = 15 T=15 . Each y t y_{t} is decoded back to a frame by k k -nearest-neighbor retrieval ( k = 1 k=1 , Euclidean distance) against the library’s embeddings, yielding a decoded joint-space path { θ ^ t } \{\hat{\theta}_{t}\} . We evaluate this path against the ideal straight-in-joint-space trajectory connecting θ s \theta_{s} and θ g \theta_{g} .

##### Metric.

We sample K = 30 K=30 start–goal pairs uniformly from the eval library (shared seed across encoders) and report path length : ( ∑ t ‖ θ ^ t + 1 − θ ^ t ‖ ) / ‖ θ g − θ s ‖ \big(\sum_{t}\|\hat{\theta}_{t+1}-\hat{\theta}_{t}\|\big)/\|\theta_{g}-\theta_{s}\| , the θ \theta -space arc length of the decoded path divided by the chord length. Ideal value 1 1 (straight chord); larger values indicate the encoder warps joint-space geometry. We also track mean orthogonal deviation from the chord as a complementary diagnostic.

##### Interpretation.

Under an affine encoder h ⁡ ( z ) = Q ​ z h(z)=Qz , the latent straight line maps to the true straight line in joint space and the metric attains its ideal value; deviations measure how much the encoder warps joint-space geometry. Fig. 17 decomposes this geometrically.

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