Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds
arXiv:2607.11069
2026
Regularization
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper gives an exact sharp isoperimetric profile for ternary martingale filtrations: among sets of fixed measure, the minimum hierarchical one-variation is the Takagi-type Bellman function T_3(x), while every n-ary set pays at least order x log(1/x). The transferable asset is an explicit occupancy-dependent lower envelope for boundary variation on a rooted n-ary hierarchy. This can become a load-aware regularizer for hierarchical MoE routing or tree-structured sparse activations, replacing blind smoothness penalties with a profile calibrated to the routed mass. The main falsifiable benefit is more coherent and stable routing at equal load balance and task loss.
Ideas from this paper
Unverified
2026
Represent MoE experts as leaves of a balanced ternary tree and regularize the hierarchical boundary of each expert's assignment mask. At fixed routing mass x, the ternary martingale isoperimetric theorem supplies the explicit minimum one-variation T_3(x), so the router can be penalized according to an occupancy-dependent profile rather than a uniform parent-child disagreement cost. This should favor coherent, stable routing regions while preventing small expert supports from obtaining…
Useful5/10
Difficulty5/10
Novelty7/10