File size: 2,907 Bytes
b317718 | 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 77 78 79 80 81 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace CurrentProjection
abbrev Vector (d : ℕ) := EuclideanSpace ℝ (Fin d)
abbrev Sphere (d : ℕ) := Metric.sphere (0 : Vector d) 1
abbrev Rows (k d : ℕ) := Fin k → Vector d
def uniformSphere (d : ℕ) : Measure (Sphere d) :=
let σ := (volume : Measure (Vector d)).toSphere
(σ Set.univ)⁻¹ • σ
def gaussianRows (k d : ℕ) : Measure (Rows k d) :=
Measure.pi fun _ => stdGaussian (Vector d)
def label {k d : ℕ} (g : Rows k d) (s : Vector d) : Fin k → ℝ :=
fun i => ⟪s, g i⟫
def LocalGrowth {d : ℕ} (ρ : Measure (Sphere d)) (a : ℕ) (B : ℝ) : Prop :=
∀ z : Vector d, ∀ t : ℝ, 0 < t →
ρ {u | dist (u : Vector d) z ≤ t} ≤ ENNReal.ofReal (B * t ^ a)
def cubeAverage {k d : ℕ} (ρ : Measure (Sphere d)) (g : Rows k d)
(y : Fin k → ℝ) (δ : ℝ) : ℝ≥0∞ :=
ENNReal.ofReal ((2 * δ)⁻¹ ^ k) *
ρ {u | ∀ i, |label g (u : Vector d) i - y i| ≤ δ}
def exactDensity {k d : ℕ} (ρ : Measure (Sphere d)) (g : Rows k d)
(y : Fin k → ℝ) : ℝ≥0∞ := by
classical
let f := fun j : ℕ => (cubeAverage ρ g y (1 / (j : ℝ))).toReal
exact if h : ∃ L : ℝ, Tendsto f atTop (𝓝 L) then ENNReal.ofReal h.choose else 0
def AdmissibleDimensions (d m r q : ℕ) : Prop :=
1 ≤ m ∧ 1 ≤ r ∧ 1 ≤ q ∧ m + r < d - 1 ∧ m + r - m - (q - 1) ≥ 1
def momentConstant (C : ℝ) (d : ℕ) (B D : ℝ) : ℝ≥0∞ :=
ENNReal.ofReal (Real.exp (C * d) * B * (1 + max (Real.log (D / B)) 0))
def MixedMomentMain : Prop :=
∃ C : ℝ, 0 < C ∧ ∀ (d m r q : ℕ), AdmissibleDimensions d m r q →
∀ (ρ : Measure (Sphere d)), IsFiniteMeasure ρ →
∀ (B D : ℝ), 0 < B → 0 < D → LocalGrowth ρ (m+r) B → LocalGrowth ρ (d-1) D →
(∀ (s : Sphere d) (δ : ℝ), 0 < δ →
(∫⁻ X, (∫⁻ Z, cubeAverage ρ (Fin.append X Z)
(label (Fin.append X Z) (s : Vector d)) δ ∂gaussianRows r d)^q
∂gaussianRows m d) ≤ (momentConstant C d B D)^q) ∧
(ρ ≪ uniformSphere d → ∃ E : Set (Sphere d), MeasurableSet E ∧ ρ E = 0 ∧
∀ s ∉ E,
(∀ᵐ G ∂gaussianRows (m+r) d,
Tendsto (fun δ : ℝ => cubeAverage ρ G (label G (s : Vector d)) δ)
(𝓝[>] 0) (𝓝 (exactDensity ρ G (label G (s : Vector d))))) ∧
((∫⁻ X, (∫⁻ Z, exactDensity ρ (Fin.append X Z)
(label (Fin.append X Z) (s : Vector d)) ∂gaussianRows r d)^q
∂gaussianRows m d) ≤ (momentConstant C d B D)^q) ∧
(ρ ≠ 0 → ∀ᵐ G ∂gaussianRows (m+r) d,
0 < exactDensity ρ G (label G (s : Vector d)) ∧
exactDensity ρ G (label G (s : Vector d)) < ∞))
end CurrentProjection
end
end OAI
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