diff --git a/docs/visualizations/roadmap.html b/docs/visualizations/roadmap.html new file mode 100644 index 0000000..06e73b8 --- /dev/null +++ b/docs/visualizations/roadmap.html @@ -0,0 +1,2311 @@ + + + + + +高维概率论 Lean 形式化路线图 + + + + + +
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高维概率论 Lean 形式化路线图

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状态图例

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节点按任务状态着色；粒度为 Lean 叶模块组 / provider contract / example surface。
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+ + + + diff --git a/docs/visualizations/roadmap.todo.md b/docs/visualizations/roadmap.todo.md new file mode 100644 index 0000000..d6bc609 --- /dev/null +++ b/docs/visualizations/roadmap.todo.md @@ -0,0 +1,214 @@ +# Theory Roadmap — Leaf-Level Formalization Tasks +> 生成于 2026-06-28 +> 共 59 个叶任务（从原 22 个概念桶细化） +> ✅ 37 已完成 • 🔄 10 进行中 • 🚫 4 阻塞 • ⏳ 8 待开始 + +## 粒度口径 + +- 一个 roadmap 条目应对应一个 Lean 叶模块组、provider contract、example/test surface，或文档化 blocker；避免用“随机矩阵”“集中不等式”这类教材章节级概念作为最小任务。 +- 状态以当前仓库文档和文件为准：已存在并被索引/测试的叶标为 `integrated`；仍是当前 RandomMatrix/Matrix Bernstein 工作面的标为 `in_progress`；文档明确不能声称已证明的 provider 缺口标为 `blocked`；后续 theorem family 标为 `pending`。 +- Git 审计结果：当前只有未跟踪文档/可视化产物：`docs/visualizations/roadmap.html`、`docs/visualizations/roadmap.todo.md`、`wiki_pages_found.md`。 + +## Depth 0 (仓库与导入面) + +- [ ] 🔄 **git-change-audit** — git 变更审计与未跟踪产物登记 + - 状态: `in_progress` / 进行中 + - 模块/文档: `docs/visualizations/roadmap.todo.md; docs/visualizations/roadmap.html; wiki_pages_found.md` +- [x] ✅ **stable-root-import** — 稳定根导入面：Init / Scalar / Statements (依赖: git-change-audit) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb.lean; HighDimProb/Init.lean; HighDimProb/Scalar.lean; HighDimProb/Statements.lean` +- [x] ✅ **experimental-aggregate** — 实验聚合导入面：Vector / Geometry / Concentration / RandomMatrix / LimitTheorems / Process (依赖: stable-root-import) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Experimental.lean` +- [x] ✅ **docs-status-indices** — 当前状态索引：Status / TODO / LeafPlan / API 文档保持短链路 (依赖: git-change-audit) + - 状态: `integrated` / 已完成 + - 模块/文档: `docs/Status.md; docs/TODO.md; docs/LeafPlan.md; docs/APIOverview.md` +- [ ] 🔄 **wiki-concept-index** — Wikipedia 概念对照表用于外部概念索引，不作为证明进度来源 (依赖: git-change-audit) + - 状态: `in_progress` / 进行中 + - 模块/文档: `wiki_pages_found.md` + +## Depth 1 (标量概率对象) + +- [x] ✅ **probability-space-events** — 概率空间、事件和可测事件对象层 (依赖: stable-root-import) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Basic.lean; HighDimProb/ProbabilitySpace.lean` +- [x] ✅ **random-variable-distribution** — 随机变量、分布和期望桥接 (依赖: probability-space-events) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomVariable.lean; HighDimProb/Distribution.lean; HighDimProb/Expectation.lean` +- [x] ✅ **tail-moment-variance-covariance** — 尾部、矩、方差、协方差基础 API (依赖: random-variable-distribution) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Tail.lean; HighDimProb/Moment.lean; HighDimProb/Scalar/Variance.lean; HighDimProb/Covariance.lean` +- [x] ✅ **lp-orlicz-scale** — Lp 与 Orlicz 标量尺度 (依赖: tail-moment-variance-covariance) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Lp.lean; HighDimProb/Orlicz.lean` +- [x] ✅ **random-family-vocabulary** — RandomFamily / Process / Sample 词汇层；不含 filtrations 或 conditioning (依赖: random-variable-distribution) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomFamily.lean; HighDimProb/RandomProcess.lean; HighDimProb/Process.lean` +- [x] ✅ **analysis-carriers** — 实不等式、自伴载体和正域分析辅助 (依赖: stable-root-import) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Analysis.lean; HighDimProb/Analysis/*.lean` + +## Depth 2 (标量集中脊柱) + +- [x] ✅ **concentration-basic-tail** — 集中基础尾部事件与通用引理 (依赖: tail-moment-variance-covariance) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Concentration/Basic.lean` +- [x] ✅ **layer-cake-markov-chebyshev** — Layer cake、Markov、Chebyshev 叶模块 (依赖: concentration-basic-tail) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Concentration/LayerCake.lean; Markov.lean; Chebyshev.lean` +- [x] ✅ **mgf-tail-cycle** — MGF 到尾部与尾部到 Orlicz/MGF 的桥接 (依赖: lp-orlicz-scale, layer-cake-markov-chebyshev) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Concentration/MGF.lean; OrliczToTail.lean; TailToOrlicz.lean` +- [x] ✅ **moment-implications** — 矩、Orlicz、固定尺度次高斯/次指数蕴含图 (依赖: mgf-tail-cycle) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Concentration/MomentImplications.lean; Implications.lean` +- [x] ✅ **subgaussian-sums** — 次高斯变量和与固定尺度矩桥 (依赖: moment-implications) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/SubGaussian.lean; HighDimProb/Concentration/SubGaussianSums.lean` +- [x] ✅ **subexponential-sums** — 次指数变量和、最大尺度和 Bernstein 原料 (依赖: moment-implications) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/SubExponential.lean; HighDimProb/Concentration/SubExponentialSums.lean; MaxScale.lean` +- [x] ✅ **rademacher-hoeffding** — Rademacher 分布/族、Rademacher 和、Hoeffding (依赖: subgaussian-sums) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Distributions/*.lean; Concentration/RademacherSums.lean; Hoeffding.lean` +- [x] ✅ **scalar-bernstein** — 标量 Bernstein min-form 与加权版本 (依赖: subexponential-sums, mgf-tail-cycle) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Concentration/Bernstein.lean` + +## Depth 3 (几何/向量/极限定理) + +- [x] ✅ **nets-metric-entropy** — Nets、覆盖/填装数、度量熵声明 (依赖: analysis-carriers) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Nets.lean; MetricEntropy.lean; MetricEntropyStatements.lean` +- [x] ✅ **gaussian-width-geometry** — 高斯宽度和几何词汇层 (依赖: nets-metric-entropy, subgaussian-sums) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/GaussianWidth.lean; HighDimProb/Geometry.lean` +- [x] ✅ **random-vector-isotropy** — 随机向量、各向同性和向量聚合 API (依赖: random-variable-distribution, lp-orlicz-scale) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomVector.lean; Isotropic.lean; Vector.lean` +- [x] ✅ **subgaussian-vector** — 次高斯随机向量接口 (依赖: random-vector-isotropy, subgaussian-sums) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/SubGaussianVector.lean` +- [x] ✅ **empirical-process-vocabulary** — 经验过程与过程侧词汇 (依赖: random-family-vocabulary, nets-metric-entropy) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/EmpiricalProcess.lean; HighDimProb/Process.lean` +- [x] ✅ **weak-law-scaffold** — 样本均值、弱大数定律和假设词汇 (依赖: random-variable-distribution, scalar-bernstein) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/LimitTheorems/*.lean` +- [x] ✅ **signal-recovery-statements** — 信号恢复/陈述层占位与术语索引 (依赖: gaussian-width-geometry, subgaussian-vector) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/SignalRecovery.lean; HighDimProb/BookStatements.lean; docs/STATEMENTS.md` + +## Depth 4 (随机矩阵基础层) + +- [x] ✅ **rm-basic-rowscols-action** — 随机矩阵基元、行列和矩阵-向量作用 (依赖: random-vector-isotropy) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/Basic.lean; RowsCols.lean; Action.lean` +- [x] ✅ **rm-norms-operatornorm-unitsphere** — 矩阵范数、算子范数和单位球面 (依赖: rm-basic-rowscols-action, nets-metric-entropy) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/Norms.lean; OperatorNorm.lean; UnitSphere.lean` +- [x] ✅ **rm-selfadjoint-order-spectral** — 自伴、PSD/Loewner 序和谱桥 (依赖: analysis-carriers, rm-norms-operatornorm-unitsphere) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/SelfAdjoint.lean; MatrixOrder.lean; Spectral.lean` +- [x] ✅ **rm-expectation-sums-vp** — 矩阵期望、有限和、方差代理 (依赖: rm-selfadjoint-order-spectral, random-family-vocabulary) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/Expectation.lean; Sums.lean; VarianceProxy.lean` +- [x] ✅ **rm-quadratic-samplecovariance** — 二次型、样本协方差和随机矩阵代数 (依赖: rm-expectation-sums-vp, random-vector-isotropy) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/QuadraticForm.lean; SampleCovariance.lean; Algebra.lean` +- [x] ✅ **rm-statement-atlas** — RandomMatrix statement atlas / hardbone typed targets (依赖: rm-quadratic-samplecovariance, scalar-bernstein) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/Statements.lean; ConcentrationStatements.lean; HardboneStatements.lean` + +## Depth 5 (TraceExp / Tropp 提供者层) + +- [x] ✅ **trace-exp-core** — TraceExp 核心 API 与 Laplace 路由 (依赖: rm-statement-atlas) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/TraceExp.lean; Laplace.lean` +- [x] ✅ **trace-exp-monotonicity** — trace-exp 导数、单调性和 provider witness (依赖: trace-exp-core, rm-selfadjoint-order-spectral) + - 状态: `integrated` / 已完成 + - 模块/文档: `TraceExpDerivative.lean; TraceExpMonotonicity.lean; TraceExpMonotonicityProvider.lean` +- [x] ✅ **cstar-log-order-transport** — CStar transport、矩阵 log provider 与 log/order bridge (依赖: trace-exp-monotonicity, analysis-carriers) + - 状态: `integrated` / 已完成 + - 模块/文档: `CStarBridge.lean; CStarOrderTransport.lean; MatrixLogProvider.lean; TraceExpLogOrderProvider.lean` +- [x] ✅ **tropp-state-bookkeeping** — prefix/suffix partial sum 和 natural trace state bookkeeping (依赖: trace-exp-core, rm-expectation-sums-vp) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/Sums.lean; TraceExpTroppStepProvider.lean` +- [ ] 🔄 **trace-mgf-consumer** — 有限族 trace-MGF consumer 与条件步骤组合 (依赖: tropp-state-bookkeeping, cstar-log-order-transport) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/TraceExp.lean; docs/STATEMENTS.md` +- [ ] 🔄 **provider-facades** — Lieb/Tropp provider-facing import layer (依赖: trace-mgf-consumer) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/LiebProvider.lean; EpsteinProvider.lean; NaturalHistoryProvider.lean` +- [x] ✅ **support-dimension-bridges** — rank/support/excess-support trace bound consumers (依赖: trace-exp-monotonicity) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/RandomMatrix/SupportProvider.lean; HardboneStatements.lean` + +## Depth 6 (矩阵 Bernstein / 样本协方差消费者) + +- [ ] 🔄 **matrix-bernstein-assumption-bundles** — CFC-free Matrix Bernstein Tropp assumption bundles (依赖: provider-facades, scalar-bernstein) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/ConcentrationStatements.lean` +- [ ] 🔄 **matrix-bernstein-tail-wrappers** — 二次型、two-sided、operator-norm Matrix Bernstein tail wrappers (依赖: matrix-bernstein-assumption-bundles, trace-mgf-consumer) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/ConcentrationStatements.lean; Laplace.lean` +- [ ] 🔄 **sample-covariance-compact-route** — 读者优先的 bounded-row sample-covariance target/record route (依赖: matrix-bernstein-tail-wrappers, rm-quadratic-samplecovariance) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/SampleCovariance.lean; ConcentrationStatements.lean` +- [ ] 🔄 **exact-row-centered-square-infra** — exact-row centered-square wrapper/bundle infrastructure (依赖: sample-covariance-compact-route) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/SampleCovariance.lean; VarianceProxy.lean` +- [ ] 🔄 **negative-side-transfer** — negative-side exact-row variance-proxy transfer adapters (依赖: exact-row-centered-square-infra) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/SampleCovariance.lean` +- [x] ✅ **reader-example-routes** — 读者级 RandomMatrix / sample covariance / feature-kernel examples (依赖: sample-covariance-compact-route) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProb/Examples/RandomMatrix/*.lean; HighDimProb/Examples/RandomMatrixUsage.lean` +- [x] ✅ **judge-api-coverage** — HighDimProbTest 与 HighDimProbJudge 覆盖 public route (依赖: reader-example-routes, matrix-bernstein-tail-wrappers) + - 状态: `integrated` / 已完成 + - 模块/文档: `HighDimProbTest/*.lean; HighDimProbJudge/**/*.lean` + +## Depth 7 (当前硬骨头提供者缺口) + +- [ ] 🚫 **epstein-lieb-concavity** — Epstein/Lieb concavity provider：当前显式假设，未证明 (依赖: provider-facades) + - 状态: `blocked` / 阻塞 + - 模块/文档: `HighDimProb/RandomMatrix/EpsteinProvider.lean; LiebProvider.lean` +- [ ] 🚫 **conditioning-independence-bridge** — 有限族 independence / conditional-expectation bridge (依赖: trace-mgf-consumer, random-family-vocabulary) + - 状态: `blocked` / 阻塞 + - 模块/文档: `HighDimProb/RandomMatrix/TraceExpTroppStepProvider.lean` +- [ ] 🚫 **trace-exp-integrability-propagation** — trace-exp integrability propagation provider (依赖: trace-exp-core, provider-facades) + - 状态: `blocked` / 阻塞 + - 模块/文档: `HighDimProb/RandomMatrix/TraceExpIntegrabilityProvider.lean` +- [ ] 🔄 **natural-history-measurability** — natural history measurability / suffix-entry provider (依赖: tropp-state-bookkeeping) + - 状态: `in_progress` / 进行中 + - 模块/文档: `HighDimProb/RandomMatrix/NaturalHistoryProvider.lean` +- [ ] 🚫 **variance-proxy-sharp-provider** — sample covariance hardbone sharp-chain provider 缺口 (依赖: exact-row-centered-square-infra) + - 状态: `blocked` / 阻塞 + - 模块/文档: `HighDimProb/RandomMatrix/VarianceProxy.lean; SampleCovariance.lean` +- [ ] ⏳ **support-provider-construction** — 非平凡 support domination / excess support construction (依赖: support-dimension-bridges) + - 状态: `pending` / 待开始 + - 模块/文档: `HighDimProb/RandomMatrix/SupportProvider.lean` +- [ ] ⏳ **tail-event-domination** — trace-MGF route 到 tail-event 的 domination 前提消解 (依赖: matrix-bernstein-tail-wrappers) + - 状态: `pending` / 待开始 + - 模块/文档: `HighDimProb/RandomMatrix/Laplace.lean; ConcentrationStatements.lean` + +## Depth 8 (后续定理族与稳定化) + +- [ ] ⏳ **full-matrix-bernstein** — 完整 Matrix Bernstein 定理：仅在 provider 缺口关闭后推进 (依赖: epstein-lieb-concavity, conditioning-independence-bridge, trace-exp-integrability-propagation, variance-proxy-sharp-provider, tail-event-domination) + - 状态: `pending` / 待开始 + - 模块/文档: `HighDimProb/RandomMatrix/ConcentrationStatements.lean` +- [ ] ⏳ **hanson-wright** — Hanson-Wright statement / proof route (依赖: full-matrix-bernstein, subgaussian-vector) + - 状态: `pending` / 待开始 + - 模块/文档: `docs/LeafPlan.md; HighDimProb/RandomMatrix/Statements.lean` +- [ ] ⏳ **covariance-estimation** — 协方差估计与 operator-norm deviation theorem family (依赖: sample-covariance-compact-route, full-matrix-bernstein) + - 状态: `pending` / 待开始 + - 模块/文档: `docs/LeafPlan.md; HighDimProb/RandomMatrix/SampleCovariance.lean` +- [ ] ⏳ **jl-rip-statements** — Johnson-Lindenstrauss / RIP / signal recovery statements (依赖: gaussian-width-geometry, covariance-estimation) + - 状态: `pending` / 待开始 + - 模块/文档: `HighDimProb/SignalRecovery.lean; docs/LeafPlan.md` +- [ ] ⏳ **strong-law-process-chaining** — SLLN、subGaussian increments 和 chaining/VC 过程族 (依赖: weak-law-scaffold, empirical-process-vocabulary) + - 状态: `pending` / 待开始 + - 模块/文档: `HighDimProb/LimitTheorems/*.lean; HighDimProb/Process.lean` +- [ ] ⏳ **stable-promotion-audit** — 实验模块升入稳定根的测试/文档/API 审计 (依赖: full-matrix-bernstein, covariance-estimation, strong-law-process-chaining) + - 状态: `pending` / 待开始 + - 模块/文档: `docs/ModuleTree.md; docs/TestPlan.md` diff --git a/wiki_pages_found.md b/wiki_pages_found.md new file mode 100644 index 0000000..08e0a76 --- /dev/null +++ b/wiki_pages_found.md @@ -0,0 +1,143 @@ +# HighDimProb Wiki 页面对照表 + +根据 `C:\Projects\HighDimProb\HighDimProb` 目录下的所有 Lean 文件中的定理/概念，找到的 Wikipedia 页面如下： + +--- + +## 根目录文件 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `Basic.lean` | 事件 (Event)、可测事件 | https://en.wikipedia.org/wiki/Probability_space (可测空间/事件) | +| `ProbabilitySpace.lean` | 概率空间 | https://en.wikipedia.org/wiki/Probability_space | +| `RandomVariable.lean` | 随机变量 | https://en.wikipedia.org/wiki/Random_variable | +| `Distribution.lean` | 概率分布 | https://en.wikipedia.org/wiki/Probability_distribution | +| `Expectation.lean` | 期望 / LOTUS | https://en.wikipedia.org/wiki/Expected_value / https://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician | +| `Tail.lean` | 尾部事件 / 尾部概率 | https://en.wikipedia.org/wiki/Fat-tailed_distribution (尾部概念) | +| `Moment.lean` | 矩 | https://en.wikipedia.org/wiki/Moment_(mathematics) | +| `Variance.lean` → `Scalar/Variance.lean` | 方差 / 协方差 | https://en.wikipedia.org/wiki/Variance / https://en.wikipedia.org/wiki/Covariance | +| `Covariance.lean` | 协方差 | https://en.wikipedia.org/wiki/Covariance | +| `Orlicz.lean` | Orlicz 函数 ψ₁, ψ₂ | https://en.wikipedia.org/wiki/Orlicz_space | +| `SubGaussian.lean` | 次高斯分布 | https://en.wikipedia.org/wiki/Sub-Gaussian_distribution | +| `SubExponential.lean` | 次指数分布 | https://en.wikipedia.org/wiki/Heavy-tailed_distribution | +| `Lp.lean` | Lᵖ 空间 | https://en.wikipedia.org/wiki/Lp_space | +| `Nets.lean` | ε-网 / 覆盖数 | https://en.wikipedia.org/wiki/Covering_number | +| `MetricEntropy.lean` | 度量熵 / 覆盖数 / 填装数 | https://en.wikipedia.org/wiki/Metric_entropy | +| `Isotropic.lean` | 各向同性随机向量 | https://en.wikipedia.org/wiki/Isotropic_vector | +| `GaussianWidth.lean` | 高斯宽度 | https://en.wikipedia.org/wiki/Gaussian_width (404 — 可能无独立页面) | +| `Geometry.lean` | 几何概念 | https://en.wikipedia.org/wiki/Convex_geometry | +| `EmpiricalProcess.lean` | 经验过程 | https://en.wikipedia.org/wiki/Empirical_process | +| `SignalRecovery.lean` | 信号恢复 | https://en.wikipedia.org/wiki/Signal_recovery | +| `RandomProcess.lean` | 随机过程 | https://en.wikipedia.org/wiki/Stochastic_process | +| `RandomVector.lean` | 随机向量 | https://en.wikipedia.org/wiki/Multivariate_random_variable | +| `SubGaussianVector.lean` | 次高斯随机向量 | https://en.wikipedia.org/wiki/Sub-Gaussian_distribution | +| `RandomMatrix.lean` | 随机矩阵 | https://en.wikipedia.org/wiki/Random_matrix | +| `Vector.lean` | 随机向量汇总 | https://en.wikipedia.org/wiki/Multivariate_random_variable | +| `Scalar.lean` | 标量概率汇总 | — (汇总页, 无独立概念) | +| `Process.lean` | 随机过程汇总 | — (汇总页) | +| `Statements.lean` | 定理陈述汇总 | — (汇总页) | +| `BookStatements.lean` | 书籍定理规格 | — (规格文件, 无独立概念) | +| `Experimental.lean` | 实验模块 | — | +| `Tactic.lean` | 策略命名空间 | — | +| `Analysis.lean` | 数学分析辅助 | https://en.wikipedia.org/wiki/Real_analysis | + +## Concentration/ 目录 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `Concentration/Basic.lean` | 尾部事件基本引理 | — (基础引理) | +| `Concentration/Markov.lean` | **Markov 不等式** | https://en.wikipedia.org/wiki/Markov_inequality | +| `Concentration/Chebyshev.lean` | **Chebyshev 不等式** | https://en.wikipedia.org/wiki/Chebyshev%27s_inequality | +| `Concentration/Hoeffding.lean` | **Hoeffding 不等式** (含 Chernoff 方法) | https://en.wikipedia.org/wiki/Hoeffding_inequality + https://en.wikipedia.org/wiki/Chernoff_bound | +| `Concentration/Bernstein.lean` | **Bernstein 不等式** (次指数集中) | https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) | +| `Concentration/MGF.lean` | 矩母函数 → 尾部 (CenteredSubGaussianMGF) | https://en.wikipedia.org/wiki/Moment-generating_function | +| `Concentration/OrliczToTail.lean` | Orlicz → 尾部不等式 | https://en.wikipedia.org/wiki/Orlicz_space | +| `Concentration/TailToOrlicz.lean` | 尾部 → Orlicz | — (证明边界) | +| `Concentration/Implications.lean` | 标量蕴含图 | — | +| `Concentration/MomentImplications.lean` | 矩蕴含关系 | — | +| `Concentration/LayerCake.lean` | 千层饼表示法 | https://en.wikipedia.org/wiki/Layer_cake_representation (或 "Layer_cake_(mathematics)" 无独立页面) | +| `Concentration/SubGaussianSums.lean` | 次高斯变量和 | https://en.wikipedia.org/wiki/Sub-Gaussian_distribution | +| `Concentration/SubExponentialSums.lean` | 次指数变量和 | https://en.wikipedia.org/wiki/Heavy-tailed_distribution | +| `Concentration/RademacherSums.lean` | Rademacher 变量加权和 | https://en.wikipedia.org/wiki/Rademacher_distribution | +| `Concentration/MaxScale.lean` | 最大尺度/方差代理 | — (辅助定义) | + +## Distributions/ 目录 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `Distributions/Rademacher.lean` | Rademacher 变量 | https://en.wikipedia.org/wiki/Rademacher_distribution | +| `Distributions/RademacherFamily.lean` | 有限 Rademacher 族 | https://en.wikipedia.org/wiki/Rademacher_distribution | + +## LimitTheorems/ 目录 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `LimitTheorems/Basic.lean` | 样本均值/样本和 | https://en.wikipedia.org/wiki/Sample_mean_and_covariance | +| `LimitTheorems/WeakLaw.lean` | **弱大数定律** | https://en.wikipedia.org/wiki/Law_of_large_numbers | +| `LimitTheorems/Assumptions.lean` | 极限定理假设 | — (假设词汇) | + +## RandomMatrix/ 目录 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `RandomMatrix/Basic.lean` | 随机矩阵基元 | https://en.wikipedia.org/wiki/Random_matrix | +| `RandomMatrix/RowsCols.lean` | 矩阵行与列 | — | +| `RandomMatrix/Action.lean` | 矩阵-向量作用 | — | +| `RandomMatrix/Norms.lean` | 矩阵范数 | https://en.wikipedia.org/wiki/Matrix_norm | +| `RandomMatrix/Assumptions.lean` | 随机矩阵假设 | — | +| `RandomMatrix/SampleCovariance.lean` | 样本协方差矩阵 | https://en.wikipedia.org/wiki/Sample_covariance_matrix | +| `RandomMatrix/QuadraticForm.lean` | 二次型 | https://en.wikipedia.org/wiki/Quadratic_form | +| `RandomMatrix/Algebra.lean` | 矩阵代数桥 | — | +| `RandomMatrix/UnitSphere.lean` | 单位球面 | https://en.wikipedia.org/wiki/Unit_sphere | +| `RandomMatrix/OperatorNorm.lean` | 算子范数 | https://en.wikipedia.org/wiki/Operator_norm | +| `RandomMatrix/SelfAdjoint.lean` | 自伴矩阵 | https://en.wikipedia.org/wiki/Self-adjoint_operator | +| `RandomMatrix/MatrixOrder.lean` | PSD / Loewner 序 | — | +| `RandomMatrix/Expectation.lean` | 矩阵逐元素期望 | — | +| `RandomMatrix/Sums.lean` | 随机矩阵有限和 | — | +| `RandomMatrix/VarianceProxy.lean` | 矩阵方差代理 | — | +| `RandomMatrix/Statements.lean` | 定理陈述层 | — | +| `RandomMatrix/ConcentrationStatements.lean` | 矩阵集中假设 | — | + +## Scalar/ 目录 + +| Lean 文件 | 核心概念/定理 | Wikipedia 页面 | +|-----------|-------------|----------------| +| `Scalar/Centering.lean` | 标量中心化 | — (中心化概念) | +| `Scalar/Variance.lean` | 标量方差 | https://en.wikipedia.org/wiki/Variance | + +## Examples/ 目录 + +| Lean 文件 | 核心概念 | Wikipedia 页面 | +|-----------|---------|----------------| +| `Examples/BasicUsage.lean` | 基本使用示例 | — | +| `Examples/NetsUsage.lean` | 网/覆盖数示例 | — | +| `Examples/OrliczUsage.lean` | Orlicz 示例 | — | +| `Examples/RandomMatrixUsage.lean` | 随机矩阵示例 | — | +| `Examples/RandomVariableUsage.lean` | 随机变量示例 | — | +| `Examples/RandomVectorUsage.lean` | 随机向量示例 | — | +| `Examples/TailUsage.lean` | 尾部不等式示例 | — | + +--- + +## 核心定理的 Wikipedia 页面总结 + +| # | 经典定理 | Lean 文件 | Wikipedia URL | +|---|---------|-----------|-------------| +| 1 | Markov 不等式 | `Concentration/Markov.lean` | https://en.wikipedia.org/wiki/Markov_inequality | +| 2 | Chebyshev 不等式 | `Concentration/Chebyshev.lean` | https://en.wikipedia.org/wiki/Chebyshev%27s_inequality | +| 3 | Hoeffding 不等式 | `Concentration/Hoeffding.lean` | https://en.wikipedia.org/wiki/Hoeffding_inequality | +| 4 | Bernstein 不等式 | `Concentration/Bernstein.lean` | https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) | +| 5 | Chernoff 界 | `Concentration/Bernstein.lean` + `Concentration/Hoeffding.lean` | https://en.wikipedia.org/wiki/Chernoff_bound | +| 6 | 弱大数定律 | `LimitTheorems/WeakLaw.lean` | https://en.wikipedia.org/wiki/Law_of_large_numbers | +| 7 | 次高斯分布 | `SubGaussian.lean`, `Concentration/SubGaussianSums.lean` | https://en.wikipedia.org/wiki/Sub-Gaussian_distribution | +| 8 | 次指数分布 | `SubExponential.lean`, `Concentration/SubExponentialSums.lean` | https://en.wikipedia.org/wiki/Heavy-tailed_distribution | +| 9 | Orlicz 空间 | `Orlicz.lean`, `Concentration/OrliczToTail.lean` | https://en.wikipedia.org/wiki/Orlicz_space | +| 10 | 矩母函数 | `Concentration/MGF.lean` | https://en.wikipedia.org/wiki/Moment-generating_function | +| 11 | Rademacher 分布 | `Distributions/Rademacher.lean` | https://en.wikipedia.org/wiki/Rademacher_distribution | +| 12 | 覆盖数 / 度量熵 | `Nets.lean`, `MetricEntropy.lean` | https://en.wikipedia.org/wiki/Covering_number / https://en.wikipedia.org/wiki/Metric_entropy | +| 13 | 集中不等式 (综述) | (全局) | https://en.wikipedia.org/wiki/Concentration_inequality | +| 14 | 随机矩阵 (综述) | `RandomMatrix/` 目录 | https://en.wikipedia.org/wiki/Random_matrix | +| 15 | 算子范数 | `RandomMatrix/OperatorNorm.lean` | https://en.wikipedia.org/wiki/Operator_norm | +| 16 | 矩阵范数 | `RandomMatrix/Norms.lean` | https://en.wikipedia.org/wiki/Matrix_norm | +| 17 | 样本协方差矩阵 | `RandomMatrix/SampleCovariance.lean` | https://en.wikipedia.org/wiki/Sample_covariance_matrix | +| 18 | 自伴算子 | `RandomMatrix/SelfAdjoint.lean` | https://en.wikipedia.org/wiki/Self-adjoint_operator |