File size: 2,141 Bytes
9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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