{
  "schema_version": "1.0",
  "title": "Reproduction: On Densest k-Subgraph Mining and Diagonal Loading (Optimization Landscape and Finite-Step Exact Convergence)",
  "emoji": "🌐",
  "space_id": "snaykey/repro-sam-saddle",
  "paper": {
    "arxiv_id": "2410.07388",
    "openreview_id": "VQt4w3lElX"
  },
  "tags": [
    "icml2026-repro",
    "paper-VQt4w3lElX"
  ],
  "updated_at": "2026-07-29T14:00:00+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: On Densest k-Subgraph Mining and Diagonal Loading (Optimization Landscape and Finite-Step Exact Convergence)",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "children": []
      },
      {
        "slug": "c1-tightness",
        "title": "Theorem 4 extends the Motzkin-Straus theorem to show the relaxed problem's maximum value equals 1+λ-1/ω, where ω is the largest clique size, establishing that diagonal loading parameter λ≥1 is the minimal value ensuring tightness of the penalty-based relaxation.",
        "children": []
      },
      {
        "slug": "c2-integral-optimum",
        "title": "Corollary 1 proves that when λ≥1 there always exists a global maximizer of the relaxed objective that is integral, so the relaxed and combinatorial problems share the same optimal value.",
        "children": []
      },
      {
        "slug": "c3-dichotomy",
        "title": "Lemma 1 shows that for λ>1 there does not exist a non-integral local maximizer, establishing the strict dichotomy that all integral stationary points are local maximizers while non-integral stationary points are strict saddles.",
        "children": []
      },
      {
        "slug": "c4-monotonicity",
        "title": "Theorem 5 shows that increasing the diagonal loading parameter beyond λ=1 introduces additional spurious local maxima, implying λ=1 optimally trades off relaxation tightness against optimization difficulty.",
        "children": []
      },
      {
        "slug": "c5-sefw",
        "title": "The saddle-escaping Frank-Wolfe algorithm (Section 3.1) has per-iteration complexity O(m+k log n) and is shown to converge in a finite number of steps to an integral local maximizer of the densest k-subgraph relaxation.",
        "children": []
      },
      {
        "slug": "c6-experiments",
        "title": "Experiments show the Frank-Wolfe algorithm achieves denser subgraphs than competing methods LRBO, L-ADMM, and EXPP while incurring lower computational cost on real-world graphs with millions of vertices (Section 4).",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "children": []
      }
    ]
  },
  "revision": 1
}