logbook.json

{
  "schema_version": "1.0",
  "title": "Reproduction: Why Deep Jacobian Spectra Separate (Depth-Induced Scaling and Singular-Vector Alignment)",
  "emoji": "📉",
  "space_id": "snaykey/repro-prob-bisection",
  "paper": {
    "arxiv_id": "2602.12384",
    "openreview_id": "2kSBDoP1rE"
  },
  "tags": [
    "icml2026-repro",
    "paper-2kSBDoP1rE"
  ],
  "updated_at": "2026-07-29T00:00:00+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: Why Deep Jacobian Spectra Separate (Depth-Induced Scaling and Singular-Vector Alignment)",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "children": [],
        "file": "pages/executive-summary/page.md"
      },
      {
        "slug": "claim-1-theorem-5-3-lyapunov",
        "title": "Theorem 5.3 gives closed-form Lyapunov exponents governing the top singular values of masked (fixed-gates) linear networks, including a finite-depth correction term (d_{i-1}-d_i)/L (Theorem 5.3).",
        "children": [],
        "file": "pages/claim-1-theorem-5-3-lyapunov/page.md"
      },
      {
        "slug": "claim-2-theorem-6-1-alignment",
        "title": "Theorem 6.1 proves that sufficiently strong spectral separation between consecutive singular values forces alignment of dominant singular vectors across the matrix product, yielding an approximately shared singular basis for intermediate Jacobians (Theorem 6.1).",
        "children": [],
        "file": "pages/claim-2-theorem-6-1-alignment/page.md"
      },
      {
        "slug": "claim-3-proposition-7-1-dynamics",
        "title": "Proposition 7.1 shows that combining depth-induced scaling and spectral separation recovers deep-linear-like singular-value training dynamics for fixed-gates networks without the balanced-initialization assumption required by prior analyses (Proposition 7.1).",
        "children": [],
        "file": "pages/claim-3-proposition-7-1-dynamics/page.md"
      },
      {
        "slug": "claim-4-proposition-4-5-arora-ode",
        "title": "Proposition 4.5 recovers the balanced deep-linear singular-value ODE of Arora et al. (2019), ds_k/dt = -L s_k(t)^{2-2/L} <grad_J L, u_k v_k^T>, as a special case of the paper's more general framework (Proposition 4.5).",
        "children": [],
        "file": "pages/claim-4-proposition-4-5-arora-ode/page.md"
      },
      {
        "slug": "claim-5-figure-1-3-rank10-training",
        "title": "On a synthetic rank-10 regression task with depth-10, width-64 networks, the top-15 Jacobian log-singular values during training converge toward the predicted theoretical Lyapunov exponents and finite-depth corrections (Figure 1, Figures 2-3).",
        "children": [],
        "file": "pages/claim-5-figure-1-3-rank10-training/page.md"
      },
      {
        "slug": "claim-6-figure-4-6-mnist-width128",
        "title": "Controlled MNIST experiments with square hidden layers of width n=128 validate the predicted emergence of depth-induced low-rank clustering and singular-vector alignment near a diagonal structure (Figure 4, Figures 5-6).",
        "children": [],
        "file": "pages/claim-6-figure-4-6-mnist-width128/page.md"
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "children": [],
        "file": "pages/conclusion/page.md"
      }
    ],
    "file": "pages/index.md"
  },
  "revision": 1
}