# Deep SE(3)-Equivariant Geometric Reasoning for Precise Placement Tasks

arXiv: https://arxiv.org/abs/2404.13478 · HTML: https://mlsanity.com/paper/2404.13478/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 20 Apr 2024.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 5 of 5)

- [Solving Rubik's Cube with a Robot Hand](https://mlsanity.com/paper/1910.07113/) (2019)
- [Plannable Approximations to MDP Homomorphisms: Equivariance under Actions](https://mlsanity.com/paper/2002.11963/) (2020)
- [StructDiffusion: Language-Guided Creation of Physically-Valid Structures using Unseen Objects](https://mlsanity.com/paper/2211.04604/) (2022)
- [Equivariant Transporter Network](https://mlsanity.com/paper/2202.09400/) (2022)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)

## Built on by (most cited first, 19 of 19)

- [Equivariant Diffusion Policy](https://mlsanity.com/paper/2407.01812/) (2024)
- [Semantic-Aware Logical Reasoning via a Semiotic Framework](https://mlsanity.com/paper/2509.24765/) (2025)
- [Logical Phase Transitions: Understanding Collapse in LLM Logical Reasoning](https://mlsanity.com/paper/2601.02902/) (2026)
- [Imagination Policy: Using Generative Point Cloud Models for Learning Manipulation Policies](https://mlsanity.com/paper/2406.11740/) (2024)
- [CordViP: Correspondence-based Visuomotor Policy for Dexterous Manipulation in Real-World](https://mlsanity.com/paper/2502.08449/) (2025)
- [FlowRAM: Grounding Flow Matching Policy with Region-Aware Mamba Framework for Robotic Manipulation](https://mlsanity.com/paper/2506.16201/) (2025)
- [AnyPlace: Learning Generalized Object Placement for Robot Manipulation](https://mlsanity.com/paper/2502.04531/) (2025)
- [Web2Grasp: Learning Functional Grasps from Web Images of Hand-Object Interactions](https://mlsanity.com/paper/2505.05517/) (2025)
- [MATCH POLICY: A Simple Pipeline from Point Cloud Registration to Manipulation Policies](https://mlsanity.com/paper/2409.15517/) (2024)
- [Constraint-Preserving Data Generation for Visuomotor Policy Learning](https://mlsanity.com/paper/2508.03944/) (2025)
- [A Practical Guide for Incorporating Symmetry in Diffusion Policy](https://mlsanity.com/paper/2505.13431/) (2025)
- [Slot-Level Robotic Placement via Visual Imitation from Single Human Video](https://mlsanity.com/paper/2504.01959/) (2025)
- [Sample-Efficient Behavior Cloning Using General Domain Knowledge](https://mlsanity.com/paper/2501.16546/) (2025)
- [AnySlot: Goal-Conditioned Vision-Language-Action Policies for Zero-Shot Slot-Level Placement](https://mlsanity.com/paper/2604.10432/) (2026)
- [Semiotic logical hexagon theory for LLM logical reasoning](https://mlsanity.com/paper/2607.21933/) (2026)
- [Beyond World-Frame Action Heads: Motion-Centric Action Frames for Vision-Language-Action Models](https://mlsanity.com/paper/2605.11809/) (2026)
- [Robustness of Robotic Manipulation: Foundations and Frontiers](https://mlsanity.com/paper/2606.31494/) (2026)
- [Action Map Policy: Learning 3D Closed-loop Manipulation via Pixel Classification](https://mlsanity.com/paper/2607.10706/) (2026)
- [Pix2Act: Image-Space Manipulation Policies with Equivariant Augmentation](https://mlsanity.com/paper/2607.11167/) (2026)

## Abstract (arXiv)

Many robot manipulation tasks can be framed as geometric reasoning tasks, where an agent must be able to precisely manipulate an object into a position that satisfies the task from a set of initial conditions. Often, task success is defined based on the relationship between two objects - for instance, hanging a mug on a rack. In such cases, the solution should be equivariant to the initial position of the objects as well as the agent, and invariant to the pose of the camera. This poses a challenge for learning systems which attempt to solve this task by learning directly from high-dimensional demonstrations: the agent must learn to be both equivariant as well as precise, which can be challenging without any inductive biases about the problem. In this work, we propose a method for precise relative pose prediction which is provably SE(3)-equivariant, can be learned from only a few demonstrations, and can generalize across variations in a class of objects. We accomplish this by factoring the problem into learning an SE(3) invariant task-specific representation of the scene and then interpreting this representation with novel geometric reasoning layers which are provably SE(3) equivariant. We demonstrate that our method can yield substantially more precise placement predictions in simulated placement tasks than previous methods trained with the same amount of data, and can accurately represent relative placement relationships data collected from real-world demonstrations. Supplementary information and videos can be found at https://sites.google.com/view/reldist-iclr-2023.