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d4317d7 | 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.MeasureTheory.Falconer.Recursion.ProfileGrid
import OAI.MeasureTheory.Falconer.Estimates.AllDimensionalOccupiedClasses
import OAI.MeasureTheory.Falconer.Estimates.RegularizationScales
namespace OAI
/-!
# Mass-profile comparison at every depth in arbitrary dimension
The interpolation loss is (d+1)*lambda. The block depths satisfy the
same numerical bracketing as in the planar case.
-/
namespace Falconer.Regularization
variable {d : ℕ}
open MeasureTheory
open scoped ENNReal
noncomputable def sourceProfile (Q : DyadicCube d) (μ : Measure (Space d))
(N B : ℕ) (p : CellPath d) : ℝ → ℝ :=
regularizedProfile (regularizationLeaves Q μ N) (regularizationLabel Q μ N B) N B (N/B+1) p
noncomputable def sourceClass (Q : DyadicCube d) (μ : Measure (Space d))
(N B : ℕ) (p : CellPath d) : Measure (Space d) :=
normalizedClass Q μ (regularizationLeaves Q μ N) (regularizationLabel Q μ N B)
(regularizationLabel Q μ N B p)
theorem sourceClass_grid_comparison (Q : DyadicCube d) (hQ : 0 < Q.side)
(μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B)
{p : CellPath d} (hp : p ∈ regularizationLeaves Q μ N)
{i : ℕ} (hi : i ≤ N/B+1) {P : CellPath d}
(hP : P ∈ (classLeaves (regularizationLeaves Q μ N) (regularizationLabel Q μ N B)
(regularizationLabel Q μ N B p)).image (List.take (blockDepth N B i))) :
(((2:ℝ)^N)^(-sourceProfile Q μ N B p (blockDepth N B i)))/(2:ℝ)^(N/B+2) ≤
(sourceClass Q μ N B p (Q.descendant P).carrier).toReal ∧
(sourceClass Q μ N B p (Q.descendant P).carrier).toReal ≤
(2:ℝ)^(N/B+2)*(((2:ℝ)^N)^(-sourceProfile Q μ N B p (blockDepth N B i))) := by
classical
rw [sourceProfile,regularizedProfile_reference_grid hN (show regularizationLeaves Q μ N ⊆ cubePaths d N from Finset.filter_subset _ _)
(regularizationLabel Q μ N B) hB p hi]
exact regularization_grid_mass_bounds Q hQ μ N hB hp hi hP
theorem profile_bracket_mass_bound {R C lam m₀ mₙ m₁ x₀ xₙ x₁ : ℝ}
(hR : 1 ≤ R) (hC : 0 < C) (hCL : C ≤ R^lam)
(hml : m₁-mₙ ≤ (d:ℝ)*lam) (hmu : mₙ-m₀ ≤ (d:ℝ)*lam)
(hlo : R^(-m₁)/C ≤ x₁) (hhi : x₀ ≤ C*R^(-m₀))
(hxl : x₁ ≤ xₙ) (hxu : xₙ ≤ x₀) :
R^(-((d:ℝ)+1)*lam)*R^(-mₙ) ≤ xₙ ∧ xₙ ≤ R^(((d:ℝ)+1)*lam)*R^(-mₙ) := by
have hRp : 0 < R := lt_of_lt_of_le zero_lt_one hR
constructor
· apply le_trans ?_ (hlo.trans hxl)
apply (le_div_iff₀ hC).mpr
calc
R^(-((d:ℝ)+1)*lam)*R^(-mₙ)*C ≤ R^(-((d:ℝ)+1)*lam)*R^(-mₙ)*R^lam :=
mul_le_mul_of_nonneg_left hCL (by positivity)
_ = R^(-(d:ℝ)*lam-mₙ) := by rw [← Real.rpow_add hRp,← Real.rpow_add hRp]; congr 1; ring
_ ≤ R^(-m₁) := Real.rpow_le_rpow_of_exponent_le hR (by linarith)
· apply le_trans (hxu.trans hhi)
calc
C*R^(-m₀) ≤ R^lam*R^(-m₀) := mul_le_mul_of_nonneg_right hCL (by positivity)
_ = R^(lam-m₀) := by rw [← Real.rpow_add hRp]; rfl
_ ≤ R^(((d:ℝ)+1)*lam-mₙ) := Real.rpow_le_rpow_of_exponent_le hR (by linarith)
_ = R^(((d:ℝ)+1)*lam)*R^(-mₙ) := Real.rpow_add hRp _ _
theorem sourceProfile_short_increment (Q : DyadicCube d) (μ : Measure (Space d))
{N B : ℕ} (hN : 0 < N) (hB : 0 < B) (p : CellPath d) {lam u v : ℝ}
(hwidth : (B:ℝ) ≤ lam*(N:ℝ)) (hu : 0 ≤ u) (huv : u ≤ v) (hv : v ≤ N)
(hshort : v-u ≤ B) :
sourceProfile Q μ N B p v-sourceProfile Q μ N B p u ≤ (d:ℝ)*lam := by
classical
have hh := regularizedProfile_increment hN (show regularizationLeaves Q μ N ⊆ cubePaths d N from Finset.filter_subset _ _)
(regularizationLabel Q μ N B) hB p hu huv hv
apply hh.2.trans
apply (div_le_iff₀ (by exact_mod_cast hN : 0 < (N:ℝ))).mpr
have hmul₁ := mul_le_mul_of_nonneg_left hwidth (Nat.cast_nonneg d)
have hmul₂ := mul_le_mul_of_nonneg_left hshort (Nat.cast_nonneg d)
nlinarith
theorem sourceClass_all_depth_comparison (Q : DyadicCube d) (hQ : 0 < Q.side)
(μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B)
{lam : ℝ} (hwidth : (B:ℝ) ≤ lam*(N:ℝ))
(hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam)
{p : CellPath d} (hp : p ∈ regularizationLeaves Q μ N)
{n : ℕ} (hn : n ≤ N) {P : CellPath d}
(hP : P ∈ (classLeaves (regularizationLeaves Q μ N) (regularizationLabel Q μ N B)
(regularizationLabel Q μ N B p)).image (List.take n)) :
((2:ℝ)^N)^(-((d:ℝ)+1)*lam)*((2:ℝ)^N)^(-sourceProfile Q μ N B p n) ≤
(sourceClass Q μ N B p (Q.descendant P).carrier).toReal ∧
(sourceClass Q μ N B p (Q.descendant P).carrier).toReal ≤
((2:ℝ)^N)^(((d:ℝ)+1)*lam)*((2:ℝ)^N)^(-sourceProfile Q μ N B p n) := by
classical
let i := (N-n)/B
have hb := PlanarFalconer.blockDepth_brackets hn hB
change i < N/B+1 ∧ blockDepth N B (i+1) ≤ n ∧ n ≤ blockDepth N B i ∧
blockDepth N B i-n ≤ B ∧ n-blockDepth N B (i+1) ≤ B at hb
obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hP
let ρ := sourceClass Q μ N B p
have : IsProbabilityMeasure ρ := regularization_class_probability Q hQ μ N B hp
have hgp := sourceClass_grid_comparison Q hQ μ hN hB hp (Nat.le_of_lt hb.1)
(Finset.mem_image.mpr ⟨q,hq,rfl⟩)
have hgm := sourceClass_grid_comparison Q hQ μ hN hB hp (by omega : i+1 ≤ N/B+1)
(Finset.mem_image.mpr ⟨q,hq,rfl⟩)
have hmono₁ : ρ (Q.descendant (q.take (blockDepth N B i))).carrier ≤
ρ (Q.descendant (q.take n)).carrier := by
apply measure_mono
have hh := descendant_subset_take Q hQ (q.take (blockDepth N B i)) n
simpa only [List.take_take,Nat.min_eq_left hb.2.2.1] using hh
have hmono₀ : ρ (Q.descendant (q.take n)).carrier ≤
ρ (Q.descendant (q.take (blockDepth N B (i+1)))).carrier := by
apply measure_mono
have hh := descendant_subset_take Q hQ (q.take n) (blockDepth N B (i+1))
simpa only [List.take_take,Nat.min_eq_left hb.2.1] using hh
have hml := sourceProfile_short_increment Q μ hN hB p hwidth
(Nat.cast_nonneg n) (Nat.cast_le.mpr hb.2.2.1)
(Nat.cast_le.mpr (Nat.sub_le N (i*B))) (by
simpa only [Nat.cast_sub hb.2.2.1] using
(show (((blockDepth N B i-n):ℕ):ℝ) ≤ (B:ℝ) by exact_mod_cast hb.2.2.2.1))
have hmu := sourceProfile_short_increment Q μ hN hB p hwidth
(Nat.cast_nonneg (blockDepth N B (i+1))) (Nat.cast_le.mpr hb.2.1) (Nat.cast_le.mpr hn) (by
simpa only [Nat.cast_sub hb.2.1] using
(show (((n-blockDepth N B (i+1)):ℕ):ℝ) ≤ (B:ℝ) by exact_mod_cast hb.2.2.2.2))
exact profile_bracket_mass_bound (one_le_pow₀ (by norm_num : (1:ℝ) ≤ 2))
(by positivity) hlarge hml hmu hgp.1 hgm.2
(ENNReal.toReal_mono (measure_ne_top ρ _) hmono₁)
(ENNReal.toReal_mono (measure_ne_top ρ _) hmono₀)
end Falconer.Regularization
end OAI
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