logbook.json

{
  "schema_version": 1,
  "title": "Reproduction: Collapsed Effective Operators for Higher-order Structures",
  "emoji": "📊",
  "space_id": "snaykey/repro-collapsed-operators",
  "paper": {
    "openreview_id": "arc2pWtZLN"
  },
  "tags": [
    "icml2026-repro",
    "paper-arc2pWtZLN"
  ],
  "updated_at": "2026-08-01T00:00:00Z",
  "root": {
    "slug": "index",
    "title": "Reproduction: Collapsed Effective Operators for Higher-order Structures",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "the-collapsed-effective-operator-s-is-defined-via-schur-complement-as-s-a-x-c-¹-xᵀ-margi",
        "title": "The Collapsed Effective Operator S is defined via Schur complement as S := A − X C⁻¹ Xᵀ, marginalizing higher-order cells onto vertices while encoding topology-mediated long-range interactions (Definition 3.3, Section 3.2).",
        "file": "pages/the-collapsed-effective-operator-s-is-defined-via-schur-complement-as-s-a-x-c-¹-xᵀ-margi/page.md",
        "children": []
      },
      {
        "slug": "the-collapsed-operator-is-spectrally-bounded-between-0-and-the-rank-0-laplacian-a-i-e-0-",
        "title": "The collapsed operator is spectrally bounded between 0 and the rank-0 Laplacian A, i.e. 0 ⪯ S ⪯ A, guaranteeing positive semi-definiteness (Proposition 3.5, Section 3.2).",
        "file": "pages/the-collapsed-operator-is-spectrally-bounded-between-0-and-the-rank-0-laplacian-a-i-e-0-/page.md",
        "children": []
      },
      {
        "slug": "eigenvalue-compression-holds-for-every-index-k-with-λ-k-s-λ-k-a-meaning-the-collapsed-op",
        "title": "Eigenvalue compression holds for every index k, with λ_k(S) ≤ λ_k(A), meaning the collapsed operator never exceeds the rank-0 Laplacian's spectrum (Corollary 3.6).",
        "file": "pages/eigenvalue-compression-holds-for-every-index-k-with-λ-k-s-λ-k-a-meaning-the-collapsed-op/page.md",
        "children": []
      },
      {
        "slug": "a-regularized-variant-s-ε-a-x-c-εi-¹xᵀ-is-introduced-with-tikhonov-regularization-to-bou",
        "title": "A regularized variant S_ε := A − X(C + εI)⁻¹Xᵀ is introduced with Tikhonov regularization to bound the collapse error while preserving efficient implicit computation (Proposition 3.10, Algorithm 1, Section 3.4).",
        "file": "pages/a-regularized-variant-s-ε-a-x-c-εi-¹xᵀ-is-introduced-with-tikhonov-regularization-to-bou/page.md",
        "children": []
      },
      {
        "slug": "the-graded-laplacian-l-remains-positive-semi-definite-only-when-the-coupling-weight-γ-k-",
        "title": "The graded Laplacian L⋆ remains positive semi-definite only when the coupling weight γ_k satisfies γ_k ≤ β_{k+1}·σ_min⁺(B_{k+1}), a condition on the boundary matrices (Proposition 3.1, Theorem 3.2, Section 3.1).",
        "file": "pages/the-graded-laplacian-l-remains-positive-semi-definite-only-when-the-coupling-weight-γ-k-/page.md",
        "children": []
      },
      {
        "slug": "spectral-clustering-using-the-collapsed-operator-improves-accuracy-from-46-9-to-70-9-on-",
        "title": "Spectral clustering using the collapsed operator improves accuracy from 46.9% to 70.9% on a protein secondary structure task compared to the baseline rank-0 Laplacian (Section 4).",
        "file": "pages/spectral-clustering-using-the-collapsed-operator-improves-accuracy-from-46-9-to-70-9-on-/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
  }
}