Canonization produces generalization bounds ranging from invariant-optimal to non-invariant depending on regularity, with Hilbert-curve ordering proven to give polynomial covering-number growth for point clouds while lexicographic sorting gives exponential growth.
defined for point clouds in Rd×n with respect to the action of translation by vectors inR d, with the metricρinduced by the Frobenius norm
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When and How to Canonize: A Generalization Perspective
Canonization produces generalization bounds ranging from invariant-optimal to non-invariant depending on regularity, with Hilbert-curve ordering proven to give polynomial covering-number growth for point clouds while lexicographic sorting gives exponential growth.