| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped BigOperators ENNReal | |
| namespace Problem358 | |
| abbrev E (d : ℕ) := EuclideanSpace ℝ (Fin d) | |
| def uniformMeasure {d : ℕ} (K : Set (E d)) : Measure (E d) := | |
| (volume K)⁻¹ • volume.restrict K | |
| def IsSupported {d : ℕ} (μ : Measure (E d)) (Y : Set (E d)) : Prop := | |
| μ (Yᶜ) = 0 | |
| def IsCoupling {d : ℕ} (μ ν : Measure (E d)) | |
| (π : Measure (E d × E d)) : Prop := | |
| IsProbabilityMeasure π ∧ | |
| Measure.map Prod.fst π = μ ∧ | |
| Measure.map Prod.snd π = ν | |
| def quadraticPlanCost {d : ℕ} (π : Measure (E d × E d)) : ℝ≥0∞ := | |
| ∫⁻ p, ENNReal.ofReal (‖p.1 - p.2‖ ^ 2) ∂π | |
| def wasserstein2 {d : ℕ} (μ ν : Measure (E d)) : ℝ := | |
| Real.sqrt | |
| (ENNReal.toReal | |
| (sInf {r : ℝ≥0∞ | | |
| ∃ π : Measure (E d × E d), | |
| IsCoupling μ ν π ∧ quadraticPlanCost π = r})) | |
| def graphPlan {d : ℕ} (ρ : Measure (E d)) (T : E d → E d) : | |
| Measure (E d × E d) := | |
| Measure.map (fun x => (x, T x)) ρ | |
| def IsQuadraticOptimalMap {d : ℕ} (ρ μ : Measure (E d)) | |
| (T : E d → E d) : Prop := | |
| Measurable T ∧ | |
| Measure.map T ρ = μ ∧ | |
| ∀ π : Measure (E d × E d), | |
| IsCoupling ρ μ π → | |
| quadraticPlanCost (graphPlan ρ T) ≤ quadraticPlanCost π | |
| def IsUniqueQuadraticOptimalMap {d : ℕ} (ρ μ : Measure (E d)) | |
| (T : E d → E d) : Prop := | |
| IsQuadraticOptimalMap ρ μ T ∧ | |
| ∀ π : Measure (E d × E d), | |
| IsCoupling ρ μ π → | |
| quadraticPlanCost π ≤ quadraticPlanCost (graphPlan ρ T) → | |
| π = graphPlan ρ T | |
| def mapL2Dist {d : ℕ} (ρ : Measure (E d)) | |
| (T S : E d → E d) : ℝ := | |
| Real.sqrt (∫ x, ‖T x - S x‖ ^ 2 ∂ρ) | |
| def cube (d : ℕ) : Set (E d) := | |
| {x | ∀ i, |x i| ≤ 1} | |
| def HasExactlyThreeAtoms {d : ℕ} (μ : Measure (E d)) | |
| (Y : Set (E d)) : Prop := | |
| ∃ y : Fin 3 → E d, Function.Injective y ∧ | |
| (∀ i, y i ∈ Y) ∧ | |
| ∃ w : Fin 3 → ℝ≥0∞, (∀ i, 0 < w i) ∧ | |
| (∑ i, w i) = 1 ∧ | |
| μ = ∑ i, (w i) • Measure.dirac (y i) | |
| end Problem358 | |
| end | |
| end OAI | |