World Modeling through Spectral Alignment

World models allow agents to plan in latent space by choosing a sequence of actions that most reduces the distance to a given goal state. Thus, planning can benefit from latent representations whose distances mirror commute-times in the environment. The spectral embedding space of the graph Laplacian provides such a representation, if it obeys a specific eigenvalue-dependent scaling. Unfortunately, instantiating the graph Laplacian is intractable in large, continuous environments. Self-supervised learning offers a natural route to such commute-time-preserving embeddings at scale. However, here we show that existing methods, which commonly encourage isotropic representations to prevent representational collapse, tend to degrade the "correct" eigenvalue-dependent scaling, leading to an inaccurate representation of commute times. To address this problem, we introduce Commute-Time-Preserving World Models (CTWMs), combining a latent displacement predictor and a log-determinant regularizer that prevents collapse, which provably recover the correctly scaled Laplacian representation under reversible deterministic dynamics and at the predictor's fixed point. In numerical simulations, CTWM matches or outperforms LeWM, a task-agnostic baseline, on several complex, continuous goal-reaching benchmarks, while using half the parameters.

Joint-embedding predictive world models support planning through latent predictions, but unconstrained joint training can collapse distinct observations to identical embeddings. Two prominent strategies for avoiding collapse are to inherit pretrained features, as in DINO-WM, or to learn representations end-to-end with anti-collapse regularization, as in LeWorldModel (LeWM). Yet avoiding collapse does not ensure that latent distances distinguish outcomes in ways that matter for the task. In object manipulation, for example, success depends on the object's position and orientation relative to the goal. Such task-relevant state information can remain accurately decodable while barely influencing latent distance. The resulting planning cost may fail to reflect how close a predicted outcome is to the task goal. In this paper, we propose SCALE (State-CAlibrated Latent Embeddings), a method that correlates sampled pairwise latent distances with distances in task-relevant state space. Added to LeWM's existing objective, SCALE preserves its architecture, requires privileged state only during training, and adds no planning-time computation. We show that SCALE improves planning success over LeWM across manipulation and navigation tasks with multiple solvers and provide a comprehensive analysis of how SCALE reshapes representation geometry to support planning.

Latent world models have become increasingly popular as a method to predict and plan in latent space rather than pixel space. Recent architectures, such as LeWorldModel (LeWM), jointly train the encoder and predictor using regularization techniques like SIGReg to prevent representation collapse. Even with such regularization preventing representation collapse, we identify a new world model failure mode of physical representation laziness, particularly noted in highly dynamic environments. For these lazy cases, the learned latent states do not collapse but nonetheless fail to represent key physical properties, causing ubiquitous downstream planning failure. To resolve this issue, we propose training-time auxiliary supervision with a lightweight "Fourier auxiliary head", which enforces physically-informed structuring of the latent space with no additional inference-time cost and can be generalized to any environment. Experimentally, we show that the auxiliary head substantially improves planning success rates in dynamic environments where the baseline LeWM exhibits physical representation laziness. It also leads to modest improvements in other environments, even when the baseline does not exhibit physical representation laziness. We further observe superior planning performance being accompanied by higher latent space correlations with key physical properties, indicating both the ability of our method to physically structure latent states and the potential planning-side benefit to the learned representation being physically structured. We also see in low-data regimes, auxiliary supervision is particularly impactful in increasing success rate. These findings support the use of our Fourier auxiliary head method to improve both overall success rate and data efficiency, while avoiding representation laziness in latent world models.