Latent Dynamics · 2026-10-02
Learning Commute-Time-Preserving World Models for Planning
Michael Hauri, Peter Buttaroni, Fabian A. Mikulasch, Friedemann Zenke
AI-assisted summary based on the paper. Consult the linked source for methods and limitations. Benchmark numbers appear after editor verification.
TL;DR
Commute-Time-Preserving World Models (CTWMs) learn an action-conditioned latent world model for planning where Euclidean distances in latent space match commute-time distances in the environment. The authors combine residual (latent displacement) prediction with a log-determinant covariance regularizer to prevent collapse while preserving the required eigenvalue-dependent scaling. They prove recovery of the correctly scaled Laplacian representation under deterministic reversible dynamics and report improved goal-reaching performance vs LeWM using half the parameters.
Why it matters
Planning in latent space requires distances that reflect temporal structure. The authors argue that common SSL collapse-avoidance methods (encouraging isotropic embeddings) can damage the eigenvalue-dependent scaling needed for commute-time preservation. CTWM targets planning geometry directly by pairing residual dynamics with a logdet regularizer, with a convergence result to the scaled Laplacian representation.
Method
- Learn encoder ϕ(s) and action-conditioned predictor T(z,a) via residual/latent displacement prediction: φ(s′)=φ(s)+T(φ(s),a); add a displacement penalty β||T(φ(s),a)||^2.
- Prevent representational collapse using a log-determinant regularizer on the latent covariance: use −(γ/2) log|Σ| where Σ≡Cov(φ(s)).
- At optimum (predictor at fixed point) under deterministic dynamics, prove encoder recovers the scaled Laplacian representation where latent squared L2 distances match commute-time distance up to a global constant.
Limitation
The main recovery guarantee assumes deterministic dynamics and that the predictor is at its fixed point; the theorem statement also relies on conditions including reversibility/deterministic setting as described in the contributions.
Abstract (from arXiv)
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.
Related papers
- ATLAS: Aligned Transport of Latent Structure for Reliable World Model Planning
- World Modeling through Spectral Alignment
- Latent World Models Should Plan with Distances Along the Latent Manifold
- How Long, Not How Close: A Learned Temporal Metric for Planning in Latent World Models
- Latent Projection for Latent Planning
Publication record
Bibliographic evidence describes where and when a record appeared. It does not validate scientific claims.
First public release: 2026-10-01 · Indexed by Pub-AI: 2026-10-02 UTC
Learning Commute-Time-Preserving World Models for Planning
Preprint posted: 2026-10-01 · Original record ↗
- arxiv
- 2610.01373
Explicit source versions (1)
- v1 · 2026-10-01
- Preprint available
Source date: 2026-10-01 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:published (official metadata)
Abstract · arxiv
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.