| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDescendants |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace CurrentFalconer.TreeND |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
| abbrev blockDepth (N B i : ℕ) : ℕ := N-i*B |
|
|
| structure RegularizedProfileData (ν : Measure (EuclideanSpace ℝ (Fin d))) |
| (N B : ℕ) (s lam : ℝ) (m : ℝ → ℝ) : Prop where |
| probability : IsProbabilityMeasure ν |
| continuous : Continuous m |
| zero : m 0 = 0 |
| slope : ∀ u v : ℝ, 0 ≤ u → u ≤ v → v ≤ N → |
| 0 ≤ m v-m u ∧ m v-m u ≤ d*(v-u)/(N:ℝ) |
| affine : ∀ i < N/B+1, ∃ a b : ℝ, ∀ u : ℝ, |
| (blockDepth N B (i+1):ℝ) ≤ u → u ≤ (blockDepth N B i:ℝ) → m u = a*u+b |
| cells : ∀ n ≤ N, ∀ P : CellPath d, P.length = n → 0 < ν (treeCell P) → |
| ((2:ℝ)^N)^(-(d+1:ℝ)*lam)*((2:ℝ)^N)^(-m n) ≤ (ν (treeCell P)).toReal ∧ |
| (ν (treeCell P)).toReal ≤ ((2:ℝ)^N)^((d+1:ℝ)*lam)*((2:ℝ)^N)^(-m n) |
| envelope : ∀ n ≤ N, s*(n:ℝ)/(N:ℝ)-(d+3:ℝ)*lam ≤ m n ∧ m n ≤ d*(n:ℝ)/(N:ℝ) |
| adjusted : ∀ u v : ℝ, 0 ≤ u → u ≤ N → 0 ≤ v → v ≤ N → |
| |(m v-(d:ℝ)/2*v/(N:ℝ))-(m u-(d:ℝ)/2*u/(N:ℝ))| ≤ (d:ℝ)/2*|v-u|/(N:ℝ) |
| packing : ∀ g p : ℕ, g ≤ p → p ≤ N → ∀ x : EuclideanSpace ℝ (Fin d), ∀ C₀ : ℝ, 0 ≤ C₀ → |
| ((occupiedCenterCells ν p x (C₀/(2:ℝ)^g)).card:ℝ) ≤ |
| (2*C₀+2*(d:ℝ)+7)^d*((2:ℝ)^N)^(m p-m g+2*(d+1:ℝ)*lam) |
|
|
| lemma RegularizedProfileData.profile_nonneg |
| {ρ : Measure (EuclideanSpace ℝ (Fin d))} {N B : ℕ} {s sig : ℝ} {m : ℝ → ℝ} |
| (h : RegularizedProfileData ρ N B s sig m) {n : ℕ} (hn : n ≤ N) : 0 ≤ m n := by |
| have hh := (h.slope 0 n le_rfl (by positivity) (by exact_mod_cast hn)).1 |
| simpa only [h.zero,sub_zero] using hh |
|
|
| lemma RegularizedProfileData.profile_le_dim |
| {ρ : Measure (EuclideanSpace ℝ (Fin d))} {N B : ℕ} {s sig : ℝ} {m : ℝ → ℝ} |
| (h : RegularizedProfileData ρ N B s sig m) (hN : 0 < N) {n : ℕ} (hn : n ≤ N) : m n ≤ d := by |
| apply (h.envelope n hn).2.trans |
| rw [div_le_iff₀ (by positivity : (0:ℝ) < N)] |
| exact mul_le_mul_of_nonneg_left (by exact_mod_cast hn) (by positivity) |
|
|
| lemma RegularizedProfileData.graph_child_lower |
| {ρ : Measure (EuclideanSpace ℝ (Fin d))} {N B : ℕ} {s sig : ℝ} {m : ℝ → ℝ} |
| (h : RegularizedProfileData ρ N B s sig m) {a p : ℕ} (hp : p ≤ N) |
| {X P : CellPath d} (hP : P ∈ graphChildren ρ a p X) : |
| ((2:ℝ)^N)^(-m p-(d+1:ℝ)*sig) ≤ ρ.real (treeCell P) := by |
| have hm := (h.cells p hp P (graphChildren_length hP) (Finset.mem_filter.mp hP).2).1 |
| rw [←Real.rpow_add (by positivity : (0:ℝ) < 2^N)] at hm |
| convert hm using 1 <;> congr 1 |
| ring |
|
|
| lemma RegularizedProfileData.graph_parent_upper |
| {ρ : Measure (EuclideanSpace ℝ (Fin d))} {N B : ℕ} {s sig : ℝ} {m : ℝ → ℝ} |
| (h : RegularizedProfileData ρ N B s sig m) {a : ℕ} (ha : a ≤ N) |
| {X : CellPath d} (hX : X ∈ dyadicPaths a) (hocc : 0 < ρ (treeCell X)) : |
| ρ.real (treeCell X) ≤ ((2:ℝ)^N)^(-m a+(d+1:ℝ)*sig) := by |
| have hm := (h.cells a ha X ((mem_dyadicPaths _ _).mp hX) hocc).2 |
| rw [←Real.rpow_add (by positivity : (0:ℝ) < 2^N)] at hm |
| convert hm using 1 <;> congr 1 |
| ring |
| end CurrentFalconer.TreeND |
| end |
| end |
|
|
| end OAI |
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|
