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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.Model
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
namespace KernelPotential
open Filter
open scoped Topology NNReal
namespace Geometry
section Cover
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
lemma card_separated_le (s : Finset E) {R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ)
(hs : ∀ c ∈ s, ‖c‖ ≤ R)
(hsep : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → δ ≤ ‖c - d‖) :
(s.card : ℝ) ≤ ((2 * R + δ) / δ) ^ Module.finrank ℝ E := by
borelize E
let μ : Measure E := Measure.addHaar
have hsmall : 0 < δ / 2 := by positivity
have hlarge : 0 < R + δ / 2 := by positivity
let A := ⋃ c ∈ s, ball c (δ / 2)
have hdisj : (s : Set E).PairwiseDisjoint (fun c => ball c (δ / 2)) := by
intro c hc d hd hcd
apply ball_disjoint_ball
rw [dist_eq_norm]
convert hsep c hc d hd hcd using 1
ring
have hsub : A ⊆ ball (0 : E) (R + δ / 2) := by
refine iUnion₂_subset fun x hx => ball_subset_ball' ?_
rw [dist_zero_right]
linarith [hs x hx]
have hb : (s.card : ℝ≥0∞) * ENNReal.ofReal ((δ / 2)^Module.finrank ℝ E) *
μ (ball 0 1) ≤ ENNReal.ofReal ((R + δ / 2)^Module.finrank ℝ E) * μ (ball 0 1) := by
calc
_ = μ A := by
rw [measure_biUnion_finset hdisj (fun _ _ => measurableSet_ball)]
simp only [μ.addHaar_ball_of_pos _ hsmall, Finset.sum_const, nsmul_eq_mul, mul_assoc]
_ ≤ μ (ball 0 (R + δ / 2)) := measure_mono hsub
_ = _ := μ.addHaar_ball_of_pos _ hlarge
have hb' := (ENNReal.mul_le_mul_iff_left (measure_ball_pos μ (0 : E) zero_lt_one).ne'
measure_ball_lt_top.ne).mp hb
have hbR : (s.card : ℝ) * (δ / 2)^Module.finrank ℝ E ≤
(R + δ / 2)^Module.finrank ℝ E := by
simpa [ENNReal.toReal_ofReal (pow_nonneg hsmall.le _)] using
ENNReal.toReal_le_of_le_ofReal (pow_nonneg hlarge.le _) hb'
calc
(s.card : ℝ) ≤ (R + δ / 2)^Module.finrank ℝ E / (δ / 2)^Module.finrank ℝ E :=
(le_div_iff₀ (pow_pos hsmall _)).mpr hbR
_ = ((2 * R + δ) / δ)^Module.finrank ℝ E := by
rw [← div_pow]
congr 1
field_simp
lemma exists_ball_cover_card_le {R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ) :
∃ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ R) ∧
(∀ x : E, ‖x‖ ≤ R → ∃ c ∈ s, dist x c ≤ δ) ∧
(s.card : ℝ) ≤ ((2 * R + δ) / δ)^Module.finrank ℝ E := by
classical
let ε : ℝ≥0 := ⟨δ, hδ.le⟩
let A := closedBall (0 : E) R
have hεpos : 0 < ε := hδ
have hε : ε ≠ 0 := ne_of_gt hεpos
obtain ⟨N, _, hNfin, hNcov⟩ := Metric.exists_finite_isCover_of_isCompact
(s := A) (ε := ε / 2) (by positivity) (isCompact_closedBall _ _)
have hp : Metric.packingNumber ε A < ⊤ := by
calc
_ = Metric.packingNumber (2 * (ε / 2)) A := by congr 1; ring
_ ≤ Metric.externalCoveringNumber (ε / 2) A :=
Metric.packingNumber_two_mul_le_externalCoveringNumber _ _
_ ≤ N.encard := hNcov.externalCoveringNumber_le_encard
_ < ⊤ := hNfin.encard_lt_top
let S := Metric.maximalSeparatedSet ε A
have hSfin : S.Finite := Set.encard_lt_top_iff.mp (by
change (Metric.maximalSeparatedSet ε A).encard < ⊤
rw [Metric.encard_maximalSeparatedSet hp.ne]
exact hp)
have hSsub : ∀ c ∈ hSfin.toFinset, ‖c‖ ≤ R := by
intro c hc
have : c ∈ S := by simpa using hc
simpa [A] using (Metric.maximalSeparatedSet_subset this)
refine ⟨hSfin.toFinset, hSsub, ?_, card_separated_le _ hR hδ hSsub ?_⟩
· intro x hx
have hcov := Metric.isCover_iff_subset_iUnion_closedBall.mp
(Metric.isCover_maximalSeparatedSet hp.ne)
have hm : x ∈ A := by simpa [A] using hx
obtain ⟨c, hc, hxc⟩ := mem_iUnion₂.mp (hcov hm)
exact ⟨c, by simpa using hc, hxc⟩
· intro c hc d hd hcd
have hs := Metric.isSeparated_maximalSeparatedSet (ε := ε) (A := A)
(by simpa using hc) (by simpa using hd) hcd
have ht := ENNReal.toReal_mono (edist_ne_top c d) hs.le
change δ ≤ (edist c d).toReal at ht
simpa [ε, edist_dist, dist_eq_norm] using ht
end Cover
section PlaneCover
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
lemma exists_affine_tube_cover (V : AffineSubspace ℝ E) (hV : (V : Set E).Nonempty)
{R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ) :
∃ s : Finset E,
(∀ x : E, ‖x‖ ≤ R → infDist x (V : Set E) ≤ δ → ∃ c ∈ s, dist x c ≤ 2 * δ) ∧
(s.card : ℝ) ≤ ((4 * R + 5 * δ) / δ)^Module.finrank ℝ V.direction := by
classical
by_cases hempty : {x : E | ‖x‖ ≤ R ∧ infDist x (V : Set E) ≤ δ}.Nonempty
· obtain ⟨x₀, hx₀n, hx₀d⟩ := hempty
obtain ⟨v₀, hv₀V, hv₀d⟩ := V.closed_of_finiteDimensional.exists_infDist_eq_dist hV x₀
have hn₀ : ‖v₀‖ ≤ R + δ := by
have ht := dist_triangle v₀ x₀ 0
rw [dist_zero_right, dist_zero_right, dist_comm v₀ x₀, ← hv₀d] at ht
linarith
obtain ⟨s, _, hcover, hcard⟩ := exists_ball_cover_card_le (E := V.direction)
(R := 2 * (R + δ)) (δ := δ) (by positivity) hδ
let f : V.direction ↪ E := ⟨fun v => (v : E) + v₀, by
intro v w h
exact Subtype.ext (add_right_cancel h)⟩
refine ⟨s.map f, ?_, ?_⟩
· intro x hxn hxd
obtain ⟨v, hvV, hvd⟩ := V.closed_of_finiteDimensional.exists_infDist_eq_dist hV x
have hnv : ‖v‖ ≤ R + δ := by
have ht := dist_triangle v x 0
rw [dist_zero_right, dist_zero_right, dist_comm v x, ← hvd] at ht
linarith
let u : V.direction := ⟨v - v₀, V.vsub_mem_direction hvV hv₀V⟩
have hnu : ‖u‖ ≤ 2 * (R + δ) := by
change ‖v - v₀‖ ≤ _
exact (norm_sub_le v v₀).trans (by linarith)
obtain ⟨a, ha, hu⟩ := hcover u hnu
refine ⟨f a, Finset.mem_map.mpr ⟨a, ha, rfl⟩, ?_⟩
have he : dist v (f a) = dist u a := by
rw [dist_eq_norm, dist_eq_norm]
change ‖v - ((a : E) + v₀)‖ = ‖(v - v₀) - (a : E)‖
congr 1
abel
calc
dist x (f a) ≤ dist x v + dist v (f a) := dist_triangle _ _ _
_ ≤ δ + δ := add_le_add (hvd ▸ hxd) (he ▸ hu)
_ = 2 * δ := by ring
· simpa only [Finset.card_map, show 2 * (2 * (R + δ)) + δ = 4 * R + 5 * δ by ring] using hcard
· refine ⟨∅, ?_, ?_⟩
· intro x hxn hxd
exact (hempty ⟨x, hxn, hxd⟩).elim
· simp only [Finset.card_empty, Nat.cast_zero]
positivity
end PlaneCover
end Geometry
end KernelPotential
end OAI
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