Tube-volume bounds for smooth Pfaffian hypersurfaces yield condition-number tails for Pfaffian neural classifiers, with polynomial-in-width control for single-layer rational-weight sigmoids.
Generalizations of Joints Problem
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at which three $2$-planes intersect and span $\mathbb{R}^6$ is at most $CN^{3/2+\epsilon}$. We also get almost sharp bound on the number of joints with multiplicities. The main tools are polynomial partitioning and induction on dimension.
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Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks
Tube-volume bounds for smooth Pfaffian hypersurfaces yield condition-number tails for Pfaffian neural classifiers, with polynomial-in-width control for single-layer rational-weight sigmoids.