幾何代數轉換器

摘要

幾何數據相關問題在各個領域中出現，包括計算機視覺、機器人技術、化學和物理學。這類數據可以採取多種形式，如點、方向向量、平面或轉換，但迄今為止還沒有一種單一架構可以應用於如此廣泛的幾何類型並尊重它們的對稱性。本文介紹了幾何代數變換器（GATr），這是一種通用的幾何數據架構。GATr在射影幾何代數中表示輸入、輸出和隱藏狀態，該代數提供了常見幾何對象的高效16維向量空間表示，以及作用於它們的運算符。GATr對於3D歐幾里德空間的對稱群E(3)是等變的。作為一種變換器，GATr具有可擴展性、表達力和多功能性。在n體建模和機器人規劃的實驗中，GATr表現出比非幾何基線更強大的改進。

English

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

幾何代數轉換器

Geometric Algebra Transformers

摘要

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