| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Measures.RegularizationLabelCount |
|
|
| namespace OAI |
|
|
| /-! Quantitative estimates for the distance-set construction. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| noncomputable def treeCenter (p : CellPath d) : EuclideanSpace ℝ (Fin d) := |
| ambientGridCenter p.length (fun i => (pathIndex p i : ℤ)) |
|
|
| lemma treeCenter_mem (p : CellPath d) : treeCenter p ∈ treeCell p := |
| ambientGridCenter_mem _ _ |
|
|
| noncomputable def occupiedCenterCells (ν : Measure (EuclideanSpace ℝ (Fin d))) |
| (p : ℕ) (x : EuclideanSpace ℝ (Fin d)) (r : ℝ) : Finset (CellPath d) := by |
| classical |
| exact (dyadicPaths p).filter fun q => 0 < ν (treeCell q) ∧ treeCenter q ∈ ball x r |
|
|
| lemma treeCenter_ball_card {g : ℕ} (T : Finset (CellPath d)) (hT : T ⊆ dyadicPaths g) |
| (x : EuclideanSpace ℝ (Fin d)) {r : ℝ} (hr : 0 ≤ r) |
| (hdist : ∀ p ∈ T, dist (treeCenter p) x ≤ r) : |
| (T.card : ℝ) ≤ (2*(2:ℝ)^g*r+5)^d := by |
| classical |
| let f : CellPath d → (Fin d → ℤ) := fun p i => pathIndex p i |
| have hf : Set.InjOn f T := by |
| intro p hp q hq he |
| apply pathIndex_injective (((mem_dyadicPaths _ _).mp (hT hp)).trans |
| ((mem_dyadicPaths _ _).mp (hT hq)).symm) |
| funext i |
| have hei : (pathIndex p i : ℤ) = pathIndex q i := congrFun he i |
| exact_mod_cast hei |
| have hh := ambientGridCenter_ball_card g (T.image f) x hr (fun a ha => by |
| obtain ⟨p,hp,rfl⟩ := Finset.mem_image.mp ha |
| have he : p.length = g := (mem_dyadicPaths _ _).mp (hT hp) |
| simpa only [treeCenter,he] using hdist p hp) |
| rwa [Finset.card_image_of_injOn hf] at hh |
|
|
| theorem occupied_center_count_le (ν : Measure (EuclideanSpace ℝ (Fin d))) |
| [IsFiniteMeasure ν] {g p : ℕ} (hgp : g ≤ p) |
| (x : EuclideanSpace ℝ (Fin d)) {r C₀ lo hi : ℝ} (hC₀ : 0 ≤ C₀) |
| (hr : r ≤ C₀/(2:ℝ)^g) (hlo : 0 < lo) (hhi : 0 ≤ hi) |
| (hfine : ∀ q ∈ dyadicPaths p, 0 < ν (treeCell q) → lo ≤ (ν (treeCell q)).toReal) |
| (hcoarse : ∀ q ∈ dyadicPaths g, 0 < ν (treeCell q) → (ν (treeCell q)).toReal ≤ hi) : |
| ((occupiedCenterCells ν p x r).card:ℝ) ≤ (2*C₀+2*(d:ℝ)+7)^d*hi/lo := by |
| classical |
| let S := occupiedCenterCells ν p x r |
| let T := S.image (List.take g) |
| have hS : S ⊆ dyadicPaths p := Finset.filter_subset _ _ |
| have hT : T ⊆ dyadicPaths g := by |
| intro c hc |
| obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hc |
| have hl := (mem_dyadicPaths _ _).mp (hS hq) |
| simp [mem_dyadicPaths,List.length_take,hl,hgp] |
| have hpos {q : CellPath d} (hq : q ∈ S) : 0 < ν (treeCell q) := |
| (Finset.mem_filter.mp hq).2.1 |
| have hparent (q : CellPath d) : treeCell q ⊆ treeCell (q.take g) := treeCell_subset_take q g |
| have hcard : (T.card:ℝ) ≤ (2*C₀+2*(d:ℝ)+7)^d := by |
| have hh := treeCenter_ball_card T hT x (r := (C₀+(d+1:ℝ))/(2:ℝ)^g) (by positivity) (by |
| intro c hc |
| obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hc |
| have hlen : (q.take g).length = g := (mem_dyadicPaths _ _).mp |
| (hT (Finset.mem_image.mpr ⟨q,hq,rfl⟩)) |
| have hcell := ambientGridCell_dist_le (hparent q (treeCenter_mem q)) (treeCenter_mem (q.take g)) |
| change dist (treeCenter (q.take g)) (treeCenter q) ≤ (d+1:ℝ)/(2:ℝ)^(q.take g).length at hcell |
| rw [hlen] at hcell |
| have hx := (Finset.mem_filter.mp hq).2.2 |
| have hdist : dist (treeCenter q) x ≤ C₀/(2:ℝ)^g := (mem_ball.mp hx).le.trans hr |
| calc |
| _ ≤ dist (treeCenter (q.take g)) (treeCenter q)+dist (treeCenter q) x := dist_triangle _ _ _ |
| _ ≤ (d+1:ℝ)/(2:ℝ)^g+C₀/(2:ℝ)^g := add_le_add hcell hdist |
| _ = _ := by ring) |
| have he : 2*(2:ℝ)^g*((C₀+(d+1:ℝ))/(2:ℝ)^g)+5 = 2*C₀+2*(d:ℝ)+7 := by |
| field_simp |
| ring |
| simpa only [he] using hh |
| have hcover : leafUnion S ⊆ leafUnion T := by |
| intro z hz |
| obtain ⟨q,hq,hz⟩ := mem_iUnion₂.mp hz |
| exact mem_iUnion₂.mpr ⟨q.take g,Finset.mem_image.mpr ⟨q,hq,rfl⟩,hparent q hz⟩ |
| have hsum : (S.card:ℝ)*lo ≤ (T.card:ℝ)*hi := by |
| calc |
| _ = ∑ _q ∈ S, lo := by simp |
| _ ≤ ∑ q ∈ S, (ν (treeCell q)).toReal := |
| Finset.sum_le_sum (fun q hq => hfine q (hS hq) (hpos hq)) |
| _ = (ν (leafUnion S)).toReal := (restriction_total_mass_real ν hS).symm |
| _ ≤ (ν (leafUnion T)).toReal := ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono hcover) |
| _ = ∑ q ∈ T, (ν (treeCell q)).toReal := restriction_total_mass_real ν hT |
| _ ≤ ∑ _q ∈ T, hi := by |
| apply Finset.sum_le_sum |
| intro c hc |
| apply hcoarse c (hT hc) |
| obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hc |
| exact (hpos hq).trans_le (measure_mono (hparent q)) |
| _ = _ := by simp |
| exact (le_div_iff₀ hlo).mpr (hsum.trans (mul_le_mul_of_nonneg_right hcard hhi)) |
|
|
| theorem sourceClass_local_count (μ : Measure (EuclideanSpace ℝ (Fin 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) |
| {p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves μ N) |
| {g p : ℕ} (hgp : g ≤ p) (hpN : p ≤ N) (x : EuclideanSpace ℝ (Fin d)) |
| {C₀ : ℝ} (hC₀ : 0 ≤ C₀) : |
| ((occupiedCenterCells (sourceClass μ N B p₀) p x (C₀/(2:ℝ)^g)).card:ℝ) ≤ |
| (2*C₀+2*(d:ℝ)+7)^d*((2:ℝ)^N)^ |
| (sourceProfile μ N B p₀ p-sourceProfile μ N B p₀ g+2*(d+1:ℝ)*lam) := by |
| have : IsProbabilityMeasure (sourceClass μ N B p₀) := sourceClass_probability μ N B hp₀ |
| let R : ℝ := (2:ℝ)^N |
| have hRp : 0 < R := by dsimp [R]; positivity |
| have hh := occupied_center_count_le (sourceClass μ N B p₀) hgp x hC₀ le_rfl |
| (lo := R^(-(d+1:ℝ)*lam)*R^(-sourceProfile μ N B p₀ p)) (by positivity) |
| (hi := R^((d+1:ℝ)*lam)*R^(-sourceProfile μ N B p₀ g)) (by positivity) |
| (fun q hq hpos => (sourceClass_occupied_comparison μ hN hB hwidth hlarge hp₀ hpN |
| ((mem_dyadicPaths p q).mp hq) hpos).1) |
| (fun q hq hpos => (sourceClass_occupied_comparison μ hN hB hwidth hlarge hp₀ (hgp.trans hpN) |
| ((mem_dyadicPaths g q).mp hq) hpos).2) |
| apply hh.trans_eq |
| rw [mul_div_assoc] |
| congr 1 |
| rw [← Real.rpow_add hRp,← Real.rpow_add hRp,← Real.rpow_sub hRp] |
| congr 1 |
| ring |
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| 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) |
|
|
| theorem actual_profile_regularization (μ : Measure (EuclideanSpace ℝ (Fin d))) |
| [IsProbabilityMeasure μ] (hcarry : μ (treeCell [])ᶜ = 0) |
| {s C lam : ℝ} (hC : 0 ≤ C) (hlam : 0 < lam) |
| (hball : ∀ x : EuclideanSpace ℝ (Fin d), ∀ r > 0, μ (ball x r) ≤ ENNReal.ofReal (C*r^s)) : |
| ∃ H N₀ : ℕ, 0 < H ∧ (H:ℝ) ≤ 1/lam+2 ∧ ∀ N ≥ N₀, |
| let B := N/H |
| let t := ENNReal.ofReal (((2:ℝ)^N)^(-lam)) |
| let L := regularizationLeaves μ N |
| let F := regularizationLabel μ N B |
| 0 < N ∧ 0 < B ∧ |
| (retainedClassLabels μ N B t).card ≤ ((d+2)*N+1)^(2*H+1) ∧ |
| μ (retainedClassUnion μ N B t)ᶜ ≤ ENNReal.ofReal (((2:ℝ)^N)^(-2:ℝ))+ |
| ((((d+2)*N+1)^(2*H+1):ℕ):ENNReal)*t ∧ |
| ∀ T ∈ retainedClassLabels μ N B t, |
| t ≤ classMeasure μ L F T univ ∧ |
| ∃ m : ℝ → ℝ, RegularizedProfileData (normalizedClass μ L F T) N B s lam m := by |
| classical |
| obtain ⟨H,N₁,hH,hHbound,hscales⟩ := exists_bounded_regularization_control hlam |
| obtain ⟨N₂,hconst⟩ := fixed_constant_absorbed hlam (C*(d+2:ℝ)^s) |
| refine ⟨H,max H (max N₁ N₂),hH,hHbound,?_⟩ |
| intro N hN |
| dsimp only |
| have hNH : H ≤ N := (le_max_left _ _).trans hN |
| have hN₁ : N₁ ≤ N := (le_max_left _ _).trans ((le_max_right _ _).trans hN) |
| have hN₂ : N₂ ≤ N := (le_max_right _ _).trans ((le_max_right _ _).trans hN) |
| have hs := hscales N hN₁ |
| have hc : (d*N+1)^(N/(N/H)+1)*((d+2)*N+1) ≤ ((d+2)*N+1)^(2*H+1) := by |
| rw [pow_succ ((d+2)*N+1) (2*H)] |
| apply Nat.mul_le_mul_right |
| exact (Nat.pow_le_pow_left (by nlinarith) _).trans (Nat.pow_le_pow_right (by omega) hs.2.2.1) |
| refine ⟨hs.1,hs.2.1,(retainedClassLabels_count μ N (N/H) _).trans hc,?_,?_⟩ |
| · have hh := retainedClass_discard_mass μ hcarry N (N/H) |
| (ENNReal.ofReal (((2:ℝ)^N)^(-lam))) |
| rw [thin_leaf_loss_identity] at hh |
| apply hh.trans |
| apply add_le_add_right |
| apply mul_le_mul_left |
| exact_mod_cast hc |
| · intro T hT |
| refine ⟨retainedClass_mass_lower μ hT,?_⟩ |
| obtain ⟨p,hp,he⟩ := Finset.mem_image.mp (Finset.mem_filter.mp hT).1 |
| subst T |
| refine ⟨sourceProfile μ N (N/H) p,?_⟩ |
| refine ⟨sourceClass_probability μ N (N/H) hp, |
| regularizedProfile_continuous _ _ _ _ _ _,regularizedProfile_zero _ _ _ _ _ _, |
| ?_,?_,?_,?_,?_,?_⟩ |
| · intro u v hu huv hv |
| exact regularizedProfile_increment hs.1 |
| (show regularizationLeaves μ N ⊆ dyadicPaths N from Finset.filter_subset _ _) |
| (regularizationLabel μ N (N/H)) hs.2.1 p hu huv hv |
| · exact fun i hi => sourceProfile_affine_on_blocks μ N (N/H) p hi |
| · intro n hn P hP hpos |
| exact sourceClass_occupied_comparison μ hs.1 hs.2.1 hs.2.2.2.1 hs.2.2.2.2 hp hn hP hpos |
| · intro n hn |
| refine ⟨?_,sourceProfile_upper μ hs.1 hs.2.1 p (Nat.cast_nonneg n) (Nat.cast_le.mpr hn)⟩ |
| apply sourceProfile_frostman_lower μ hs.1 hs.2.1 hs.2.2.2.1 hs.2.2.2.2 hp |
| (retained_label_real_mass_lower μ hT) _ hn |
| intro k hk P hP |
| have hh := dyadic_cell_frostman_profile_bound μ hC hball hs.1 P |
| rw [hP] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_right (hconst N hN₂) (by positivity)) |
| · exact fun u v hu huN hv hvN => sourceProfile_adjusted_lipschitz μ hs.1 hs.2.1 p hu huN hv hvN |
| · exact fun g p' hgp hpN x C₀ hC₀ => |
| sourceClass_local_count μ hs.1 hs.2.1 hs.2.2.2.1 hs.2.2.2.2 hp hgp hpN x hC₀ |
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
