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Abstract We derive sparse bounds for the bilinear spherical maximal function in any dimension . When , this immediately recovers the sharp bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity improving estimates for the single‐scale bilinear spherical averaging operator.
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Equivariant frames and the impossibility of continuous canonicalization
Canonicalization provides an architecture-agnostic method for enforcing equivariance, with generalizations such as frame-averaging recently gaining prominence as a lightweight and flexible alternative to equivariant architectures. Recent works have found an empirical benefit to using probabilistic frames instead, which learn weighted distributions over group elements. In this work, we provide strong theoretical justification for this phenomenon: for commonly-used groups, there is no efficiently computable choice of frame that preserves continuity of the function being averaged. In other words, unweighted frame-averaging can turn a smooth, non-symmetric function into a discontinuous, symmetric function. To address this fundamental robustness problem, we formally define and construct weighted frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of SO(d), O(d), and Sn on point clouds.
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- PAR ID:
- 10621769
- Publisher / Repository:
- ICML'24: Proceedings of the 41st International Conference on Machine Learning
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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