{
 "artifacts": null,
 "category": "architecture",
 "description": "Prune higher-order tensor or polynomial interactions while preserving an exchange-closed support instead of independently retaining the largest weights. The resulting sparse interaction pattern avoids arbitrary holes and retains structured substitutions between coordinates, potentially improving parameter efficiency and robustness after pruning.",
 "formulas_latex": [
  "$$S=\\{\\alpha\\in\\mathbb{Z}_{\\ge 0}^{n}:|\\alpha|=d,\\ c_{\\alpha}\u003e0\\}.$$",
  "$$\\alpha,\\beta\\in S,\\ \\alpha_i\u003e\\beta_i\\ \\Longrightarrow\\ \\exists j\\text{ with }\\alpha_j\u003c\\beta_j:\\ \\alpha-e_i+e_j\\in S\\ \\text{and}\\ \\beta+e_i-e_j\\in S.$$",
  "$$\\widetilde{f}_{S}(x)=\\sum_{\\alpha\\in S}w_{\\alpha}\\frac{x^{\\alpha}}{\\alpha!},\\qquad |S|\\ll {n+d-1\\choose d}.$$",
  "$$S_{k+1}=\\operatorname{Repair}_{\\mathrm{exchange}}\\left(S_k\\setminus\\{\\alpha_{\\min}\\}\\cup\\{\\alpha_{\\mathrm{new}}\\}\\right).$$"
 ],
 "id": 3090,
 "implementation": "Apply this to a tensorized MLP or polynomial feature layer. Represent an interaction index by alpha=(alpha_1,...,alpha_n) with nonnegative integer entries summing to degree d. The dense candidate layer computes phi_alpha(h), such as products of projected features, and has one parameter w_alpha per alpha. Begin with a support S containing the largest-magnitude or largest-gradient interactions. Every pruning round scores active interactions using running |w_alpha|, |w_alpha times gradient_alpha|, or a validation sensitivity estimate. Remove the lowest-score alpha and propose a replacement alpha_new from the inactive set. Repair the support using the exchange rule: for each pair alpha,beta in the proposed support and each i with alpha_i\u003ebeta_i, search for j with alpha_j\u003cbeta_j; if alpha-e_i+e_j or beta+e_i-e_j is absent, insert the missing feasible neighbor with the highest score, or reject the removal if no neighbor exists. After repair, the forward computation is f_S(h)=sum_{alpha in S} w_alpha phi_alpha(h)/alpha!, with the mask fixed between pruning rounds. Directly check the exchange axiom on small supports; for larger supports, check only sampled pairs and maintain a bounded exchange-neighborhood cache. Start with n=16,d=3 and retain 10%, 25%, and 50% of interactions in a CIFAR-100 tensorized MLP or a small Transformer feed-forward block. Compare against magnitude pruning, random pruning, and unstructured sparse training at equal parameter count and retraining budget. Measure validation accuracy or perplexity, activation outliers, gradient variance, and actual wall-clock memory. The target signal is equal accuracy at lower parameter count, smoother loss immediately after pruning, and improved hardware sparsity; a 2x parameter reduction at iso-accuracy is a meaningful positive result.",
 "math_summary": "For a homogeneous degree-d polynomial, the support is S={alpha in Z_+^n: |alpha|=d and c_alpha\u003e0}. The paper establishes discrete-convex support structure, including polymatroid-base behavior and generalized-polymatroid Newton polytopes. The key exchange condition is: for alpha,beta in S and any i with alpha_i\u003ebeta_i, there exists j with alpha_j\u003cbeta_j such that alpha-e_i+e_j and beta+e_i-e_j also belong to S. Here e_i is the i-th standard basis vector. Thus moving one unit of degree from coordinate i to coordinate j must be possible simultaneously for the pair of support elements. A neural interaction layer uses the masked divided-power polynomial f_S(x)=sum_{alpha in S} w_alpha x^alpha/alpha!, where S is maintained as an exchange-feasible sparse support. This is a concrete support constraint inspired by the paper's M^natural-convex and algebraic-polymatroid conclusions.",
 "math_tags": [
  "combinatorics",
  "convex-analysis",
  "geometry"
 ],
 "ml_areas": [
  "mlp",
  "pruning",
  "memory"
 ],
 "paper": {
  "arxiv_id": "2609.02850",
  "arxiv_url": "https://arxiv.org/abs/2609.02850",
  "summary_what_math_gives_to_ml": "The paper gives constructive realizable-volume models for factorially normalized homogeneous polynomials, connecting their coefficients to Lorentzian Hessian signatures and discrete-convex supports. The transferable assets are explicit curvature inequalities and exchange-closed support structures, rather than the specific Schubert or Grothendieck applications. A practical neural adaptation is to constrain positive higher-order routing or interaction polynomials toward Lorentzian curvature, while using polymatroid-style exchange rules for structured pruning. These are most plausible in higher-order MoE routers, tensorized MLPs, and polynomial attention biases, not as replacements for standard dense Transformer blocks.",
  "title": "Canonical-row Chern flow on Bott--Samelson towers: realizable-volume models for Schubert, Grothendieck, and Lascoux polynomials",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 9,
  "usefulness": 5
 },
 "solves": [
  "memory",
  "scalability",
  "generalization"
 ],
 "title": "M-convex interaction pruning",
 "url": "https://synthcore.org/idea/3090/m-convex-interaction-pruning",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}