Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry

Key points

• We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations.1
• Using the infinitesimal differential characterization of continuous symmetries, we formulate this map as an inverse symmetry operator, analogous to an inverse PDE problem.1
• For global Lie-group symmetries, we establish a representation-theoretic population characterization: the symmetry Lie algebra is the nullspace of a positive-semidefinite operator constructed from representation coefficients and the infinitesimal group action.1
• On SO(5) benchmarks, amortized equivariant estimation of harmonic coefficients improves symmetry recovery from sparse observations, whereas ordinary least squares suffices for SL(2,R).1

Sources (1)

• [1]Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 12, 02:44 PM

  Continuous symmetry discovery is typically formulated as a separate estimation problem for each function or dataset.

  We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations.

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