Tropical Circuits with Scalar Multiplication Gates
arXiv:2607.11540
2026
Architecture
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper gives an exact duality between positively homogeneous convex piecewise-linear functions and Newton polytopes: addition, maximum, and positive scalar multiplication of functions correspond respectively to Minkowski sum, convex-hull union, and dilation of polytopes. Its trace lemma further states that a max-plus circuit computes the support function of the convex hull of all coefficient vectors generated by its traces. This provides a concrete geometric parameterization for monotone or input-convex networks, together with a way to measure expressivity through the generated polytope rather than only through neuron count. The most promising transfer is a polytope-structured convex network with explicit Minkowski/convex-hull composition and a trace-geometry regularizer.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Build a positively homogeneous convex network by representing every intermediate unit as a compact polytope and composing units with Minkowski sums, convex-hull unions, and positive dilations. This gives an explicitly convex and monotone architecture whose geometric complexity can be controlled independently of the number of sampled linear pieces, potentially producing smaller ICNNs for structured convex functions.
Useful7/10
Difficulty7/10
Novelty7/10
Unverified
2026
Use the trace representation of a maxout network to regularize the geometry of its generated coefficient vectors. Encourage active traces to be diverse and nonredundant, so the model spends parameters on genuinely different supporting hyperplanes rather than branches that collapse to the same linear function.
Useful6/10
Difficulty5/10
Novelty8/10