MoE Top-k routing equals the k-th elementary symmetric tropical polynomial, making sparsity combinatorial depth that scales capacity by binom(N,k) and gives MoE combinatorial resilience on manifolds.
Tropnnc: Structured neural network compression using tropical geometry.arXiv preprint arXiv:2409.03945
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The paper claims transformer expressivity is governed by power-diagram partitions, with a tight Θ(N^{min{H,d-1}L}) linear-region bound, but the proof's multi-head vertex bound and lower-bound construction are not sound.
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Sparsity is Combinatorial Depth: Quantifying MoE Expressivity via Tropical Geometry
MoE Top-k routing equals the k-th elementary symmetric tropical polynomial, making sparsity combinatorial depth that scales capacity by binom(N,k) and gives MoE combinatorial resilience on manifolds.
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Geometric Capacity of Transformers: A Tropical Geometry Perspective
The paper claims transformer expressivity is governed by power-diagram partitions, with a tight Θ(N^{min{H,d-1}L}) linear-region bound, but the proof's multi-head vertex bound and lower-bound construction are not sound.