# Matrix-Space Reinforcement Learning for Reusing Local Transition Geometry

arXiv: https://arxiv.org/abs/2605.14304 · HTML: https://mlsanity.com/paper/2605.14304/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Deep RL algorithms and representations, one of 15 directions in Deep reinforcement learning methods.
- Builds on 30 of its 54 references that are in the map.
- First posted on arXiv 14 May 2026.

## Placed in

- Deep reinforcement learning methods / [Deep RL algorithms and representations](https://mlsanity.com/deeprl/deep-rl-algorithms-and-representations/)

## Builds on (most cited first, 30 of 30)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Progressive Neural Networks](https://mlsanity.com/paper/1606.04671/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hindsight Experience Replay](https://mlsanity.com/paper/1707.01495/) (2017)
- [When to Trust Your Model: Model-Based Policy Optimization](https://mlsanity.com/paper/1906.08253/) (2019)
- [Learning Invariant Representations for Reinforcement Learning without Reconstruction](https://mlsanity.com/paper/2006.10742/) (2020)
- [Temporal Difference Learning for Model Predictive Control](https://mlsanity.com/paper/2203.04955/) (2022)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Successor Features for Transfer in Reinforcement Learning](https://mlsanity.com/paper/1606.05312/) (2016)
- [Minigrid & Miniworld: Modular & Customizable Reinforcement Learning Environments for Goal-Oriented Tasks](https://mlsanity.com/paper/2306.13831/) (2023)
- [DeepMDP: Learning Continuous Latent Space Models for Representation Learning](https://mlsanity.com/paper/1906.02736/) (2019)
- [Scalable methods for computing state similarity in deterministic Markov Decision Processes](https://mlsanity.com/paper/1911.09291/) (2019)
- [Transfer in Deep Reinforcement Learning Using Successor Features and Generalised Policy Improvement](https://mlsanity.com/paper/1901.10964/) (2019)
- [Eigenoption Discovery through the Deep Successor Representation](https://mlsanity.com/paper/1710.11089/) (2017)
- [The Option Keyboard: Combining Skills in Reinforcement Learning](https://mlsanity.com/paper/2106.13105/) (2021)
- [Near Optimal Behavior via Approximate State Abstraction](https://mlsanity.com/paper/1701.04113/) (2017)
- [Modeling Other Players with Bayesian Beliefs for Games with Incomplete Information](https://mlsanity.com/paper/2405.14122/) (2024)
- [Network Diffuser for Placing-Scheduling Service Function Chains with Inverse Demonstration](https://mlsanity.com/paper/2501.05673/) (2025)
- [Geometry of Drifting MDPs with Path-Integral Stability Certificates](https://mlsanity.com/paper/2601.21991/) (2026)
- [Manifold-Constrained Energy-Based Transition Models for Offline Reinforcement Learning](https://mlsanity.com/paper/2602.02900/) (2026)
- [Structuring Value Representations via Geometric Coherence in Markov Decision Processes](https://mlsanity.com/paper/2602.02978/) (2026)
- [Cochain Perspectives on Temporal-Difference Signals for Learning Beyond Markov Dynamics](https://mlsanity.com/paper/2602.06939/) (2026)
- [NonZero: Interaction-Guided Exploration for Multi-Agent Monte Carlo Tree Search](https://mlsanity.com/paper/2605.00751/) (2026)
- [Operator-Guided Invariance Learning for Continuous Reinforcement Learning](https://mlsanity.com/paper/2605.06500/) (2026)

## Built on by (most cited first, 2 of 2)

- [Minimal Markovization via Stable Quotients in Holonomy-Cover Decision Processes](https://mlsanity.com/paper/2607.27132/) (2026)
- [Geometry-Aware Operator Families for Structured Representation Learning](https://mlsanity.com/paper/2609.33089/) (2026)

## Abstract (arXiv)

Compositional generalization in sequential decision-making requires identifying which parts of prior rollouts remain useful for new tasks. Existing methods reuse skills or predictive models, but often overlook rich local transition geometry and dynamics. We propose Matrix-Space Reinforcement Learning (MSRL), a geometric abstraction that represents trajectory segments through positive semidefinite matrix descriptors aggregating first- and second-order statistics of lifted one-step transitions. These descriptors expose shared hidden structure, support algebraic composition in an abstract matrix space, and reveal opportunities for transfer. We prove that the descriptor is well defined up to coordinate gauge, complete for the induced low-order additive signal class, additive under valid segment composition, and minimally sufficient among admissible additive descriptors. We further show that conditioning value functions on the trajectory-segment matrix yields a first-order smooth approximation of action values, enabling source-learned matrix-to-value mappings to bootstrap learning in new tasks. MSRL is plug-in compatible with standard model-free and model-based methods, while obstruction filtering rejects implausible compositions. Empirically, MSRL achieves the best average finite-budget target AUC of 0.73, outperforming MSRL from scratch (0.65), TD-MPC-PT+FT (0.63), and TD-MPC (0.57).