Machine learning on point clouds and other varying-size objects
summary
Defines a new invariant model on point clouds built from Gram-matrix invariants (diagonals, off‑diagonals plus one explicit polynomial f) and proves those generators yield universal approximation outside an invariant measure‑zero “bad set.” Replaces O(n^2) graph features by O(n d) complexity via rank‑d matrix completion and permutation‑invariant selection of d centers (e.g. k‑means ordered by norm), implements the generators with deep sets, shows numerical viability, and frames transferability as continuity in a limit space* that aligns task, topology, and model.
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chapters (9)
- 0:00Invariant models on point clouds
- 3:25Astrophysics and cosmology motivation
- 7:28Invariant theory and symmetries
- 13:45Constructing the machine learning model
- 21:19Transferability of varying-size models
- 31:07Compatibility and topology alignment
- 39:35Dimension-free PDE identification
- 44:46Summary and future directions
- 51:29Q&A
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