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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.Probability.GaussianRegression.Model
namespace OAI
/-!
Local mass bounds and quantitative finite Euclidean nets.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace NoiselessRegression
section FrostmanGeometry
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
def HasLocalMassBound (ν : Measure E) (a : ℝ) : Prop :=
∀ t ∈ ν.support, ∀ r : ℝ, 0 < r →
ν (Metric.closedBall t r) ≤ ENNReal.ofReal (r ^ a)
omit [BorelSpace E] in
theorem HasLocalMassBound.any_center {ν : Measure E} {a : ℝ}
(hν : HasLocalMassBound ν a) (z : E) {r : ℝ} (hr : 0 < r) :
ν (Metric.closedBall z r) ≤ ENNReal.ofReal ((2 * r) ^ a) := by
by_cases hmass : ν (Metric.closedBall z r) = 0
· simp [hmass]
obtain ⟨t, htball, htsupp⟩ := ν.nonempty_inter_support_of_pos (pos_iff_ne_zero.mpr hmass)
refine (measure_mono ?_).trans (hν t htsupp (2 * r) (by positivity))
intro x hx
apply Metric.mem_closedBall.mpr
calc
dist x t ≤ dist x z + dist z t := dist_triangle _ _ _
_ ≤ r + r := add_le_add (Metric.mem_closedBall.mp hx)
(by simpa [dist_comm] using Metric.mem_closedBall.mp htball)
_ = 2 * r := by ring
theorem card_le_volume_ratio (s : Finset E) {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R)
(hs : ∀ x ∈ s, ‖x‖ ≤ R)
(hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) :
(s.card : ℝ) ≤ ((R + r / 2) / (r / 2)) ^ Module.finrank ℝ E := by
classical
let ν : Measure E := Measure.addHaar
let δ : ℝ := r / 2
let ρ : ℝ := R + r / 2
have hδ : 0 < δ := by dsimp [δ]; positivity
have hρ : 0 < ρ := by dsimp [ρ]; positivity
let A : Set E := ⋃ c ∈ s, Metric.ball c δ
have hd : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by
intro c hc d hd hcd
apply Metric.ball_disjoint_ball
simpa [δ] using hsep c hc d hd hcd
have hsub : A ⊆ Metric.ball (0 : E) ρ := by
refine iUnion₂_subset fun x hx => ?_
apply Metric.ball_subset_ball'
dsimp [δ, ρ]
rw [dist_zero_right]
linarith [hs x hx]
have I :
(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) ≤
ENNReal.ofReal (ρ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) := by
calc
(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) = ν A := by
dsimp only [A]
rw [measure_biUnion_finset hd (fun _ _ => Metric.isOpen_ball.measurableSet)]
simp only [ν.addHaar_ball_of_pos _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc]
_ ≤ ν (Metric.ball (0 : E) ρ) := measure_mono hsub
_ = _ := ν.addHaar_ball_of_pos 0 hρ
have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) ≤
ENNReal.ofReal (ρ ^ Module.finrank ℝ E) :=
(ENNReal.mul_le_mul_iff_left (Metric.measure_ball_pos ν 0 zero_lt_one).ne'
measure_ball_lt_top.ne).mp I
have K : (s.card : ℝ) * δ ^ Module.finrank ℝ E ≤ ρ ^ Module.finrank ℝ E := by
simpa only [ENNReal.toReal_mul, ENNReal.toReal_natCast,
ENNReal.toReal_ofReal (pow_nonneg hδ.le _)] using
ENNReal.toReal_le_of_le_ofReal (pow_nonneg hρ.le _) J
rw [div_pow]
exact (le_div_iff₀ (pow_pos hδ _)).mpr K
end FrostmanGeometry
section FiniteNets
variable {E : Type*} [PseudoMetricSpace E]
theorem exists_net_of_finite_packing_bound (A : Set E) {r : ℝ} (hr : 0 ≤ r) (N : ℕ)
(hcard : ∀ s : Finset E, (∀ x ∈ s, x ∈ A) →
(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → s.card ≤ N) :
∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧
∀ x ∈ A, ∃ y ∈ s, dist x y ≤ r := by
classical
let S : Set ℕ := {n | ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ s.card = n}
have hS : S.Finite := (Set.finite_Iic N).subset (by
rintro n ⟨s, hs, hsep, rfl⟩
exact hcard s hs hsep)
have hS0 : S.Nonempty := ⟨0, ∅, by simp⟩
obtain ⟨n, hn, hmax⟩ := Set.exists_max_image S id hS hS0
obtain ⟨s, hs, hsep, rfl⟩ := hn
refine ⟨s, hs, hsep, ?_⟩
intro x hx
by_contra! hnot
have hxs : x ∉ s := by
intro h
have := hnot x h
simpa using lt_of_le_of_lt hr this
have hins : ∀ y ∈ insert x s, y ∈ A := by
intro y hy
rcases Finset.mem_insert.mp hy with rfl | hy
· exact hx
· exact hs y hy
have hsep' : ∀ y ∈ insert x s, ∀ z ∈ insert x s, y ≠ z → r < dist y z := by
intro y hy z hz hne
rcases Finset.mem_insert.mp hy with hxy | hys
· subst y
rcases Finset.mem_insert.mp hz with hxz | hzs
· exact (hne hxz.symm).elim
· exact hnot z hzs
· rcases Finset.mem_insert.mp hz with hxz | hzs
· subst z
simpa [dist_comm] using hnot y hys
· exact hsep y hys z hzs hne
have hinS : (insert x s).card ∈ S := ⟨insert x s, hins, hsep', rfl⟩
have hh := hmax _ hinS
simp only [id_eq, Finset.card_insert_of_notMem hxs] at hh
omega
end FiniteNets
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
theorem exists_unit_ball_net {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
∃ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) ∧
(s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E ∧
∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ r := by
classical
have hbound : ∀ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) →
(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) →
(s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E := by
intro s hs hsep
refine (card_le_volume_ratio s hr zero_le_one hs
(fun x hx y hy hne => (hsep x hx y hy hne).le)).trans ?_
apply pow_le_pow_left₀ (by positivity)
apply (div_le_div_iff₀ (by positivity : 0 < r / 2) hr).mpr
nlinarith
obtain ⟨s, hs, hsep, hnet⟩ := exists_net_of_finite_packing_bound
{x : E | ‖x‖ ≤ 1} hr.le ⌈(3 / r) ^ Module.finrank ℝ E⌉₊
(by
intro s hs hsep
exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _))
exact ⟨s, hs, hbound s hs hsep, hnet⟩
end NoiselessRegression
end
end OAI
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