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a2ee19a | 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 | import OAI.Probability.ProjectionMoments.WeightedCover
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace
open scoped ENNReal NNReal RealInnerProductSpace Topology
namespace ProjectionMoments
theorem positive_cap_domination :
∃ C : ℝ, 0 < C ∧ ∀ d : ℕ, 32 ≤ d →
let k := d/16
let α : ℝ := ((d:ℝ)-1)/2
∀ (c : EuclideanSpace ℝ (Fin d)), ‖c‖ = 1 →
∀ r : ℝ, 0 < r → r ≤ 2 →
∀ K : (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞,
Measurable K → (∀ z, K z ≤ 1) →
∃ D : CapDictionary c r α, ∃ a : D.I → ℝ,
(∀ i, 0 ≤ a i) ∧ Summable (fun i => a i*(r/D.radius i)^α) ∧
(∑' i, a i*(r/D.radius i)^α) ≤ Real.exp (C*d)*(capReference K c r).toReal^(1-1/(k:ℝ)) ∧
(capMeasure c r).withDensity (capKernelValue K) ≤
Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i)) ∧
(Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i))) Set.univ < ∞ := by
obtain ⟨C,hC,hdual⟩ := cap_dual_estimate
refine ⟨C,hC,?_⟩
intro d hd k α c hc r hr hr2 K hK hK1
let : NeZero d := ⟨by omega⟩
have hdR : (32:ℝ) ≤ d := by exact_mod_cast hd
have hα : 1 ≤ α := by dsimp [α]; linarith
obtain ⟨D⟩ := exists_capDictionary hc hr hα
let := D.countable
let : IsProbabilityMeasure (capMeasure c r) := capMeasure_probability hc hr hr2
let P : D.I → Measure (EuclideanSpace ℝ (Fin d)) := fun i => capMeasure (D.center i) (D.radius i)
let (i : D.I) : IsProbabilityMeasure (P i) :=
capMeasure_probability (D.center_norm i) (D.radius_pos i) ((D.radius_le i).trans hr2)
let b : D.I → ℝ := fun i => (r/D.radius i)^α
have hb : ∀ i, 1 ≤ b i := fun i => Real.one_le_rpow
((one_le_div (D.radius_pos i)).mpr (D.radius_le i)) (zero_le_one.trans hα)
let μ := (capMeasure c r).withDensity (capKernelValue K)
let : IsFiniteMeasure μ := isFiniteMeasure_withDensity (ne_of_lt
((lintegral_mono (capKernelValue_le_one hK1)).trans_lt (by simp)))
let B := Real.exp (C*d)*(capReference K c r).toReal^(1-1/(k:ℝ))
have hkR : (1:ℝ) < k := by
have hk : 2 ≤ k := by dsimp [k]; omega
exact_mod_cast (show 1 < k by omega)
have hθ : 0 ≤ 1-1/(k:ℝ) := by
have hpq := Real.HolderConjugate.conjExponent hkR
rw [one_div,hpq.one_sub_inv]
exact inv_nonneg.mpr hpq.symm.pos.le
have hqtop : capReference K c r ≠ ∞ := ne_of_lt ((capReference_le_one hK1 c r).trans_lt (by simp))
have hB : ENNReal.ofReal B = ENNReal.ofReal (Real.exp (C*d))* (capReference K c r)^(1-1/(k:ℝ)) := by
rw [ENNReal.ofReal_mul (Real.exp_pos _).le, ← ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hθ,
ENNReal.ofReal_toReal hqtop]
have htest : ∀ h : EuclideanSpace ℝ (Fin d) → ℝ≥0∞, Measurable h →
(∃ H : ℝ≥0, ∀ x, h x ≤ H) →
(∀ i, ∫⁻ x, h x ∂P i ≤ ENNReal.ofReal (b i)) → ∫⁻ x, h x ∂μ ≤ ENNReal.ofReal B := by
intro h hh _ hhD
rw [hB,lintegral_withDensity_eq_lintegral_mul _ (measurable_capKernelValue hK) hh]
simp only [Pi.mul_apply]
simpa only [mul_comm] using hdual d hd c hc r hr hr2 D K hK hK1 h hh hhD
obtain ⟨a,ha,has,hcost,hdom⟩ := Weighted.measure_domination_nonneg μ P hb D.proper
(show 0 ≤ B by positivity) htest
refine ⟨D,a,ha,has,hcost,hdom,?_⟩
rw [Measure.sum_apply _ MeasurableSet.univ]
simp only [Measure.smul_apply,measure_univ,smul_eq_mul,mul_one]
have has' : Summable a := Summable.of_nonneg_of_le ha
(fun i => by simpa using mul_le_mul_of_nonneg_left (hb i) (ha i)) has
rw [← ENNReal.ofReal_tsum_of_nonneg ha has']
exact ENNReal.ofReal_lt_top
lemma capReference_sum_le_one {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
{k : ℕ} {J : Type*} [Fintype J]
(K : J → (Fin k → E) × Label k → ℝ≥0∞) (hK : ∀ j, Measurable (K j))
(hKsum : ∀ z, ∑ j, K j z ≤ 1) (c : E) (r : ℝ) : ∑ j, capReference (K j) c r ≤ 1 := by
classical
simp only [capReference]
rw [← lintegral_finsetSum _ (fun j _ => by fun_prop)]
calc
_ = ∫⁻ X, ∫⁻ z, ∑ j, K j (X,capReferenceLabel c r X z) ∂unitBallLaw k
∂Measure.pi (fun _ => stdGaussian E) := by
apply lintegral_congr
intro X
exact (lintegral_finsetSum _ (fun j _ => by fun_prop)).symm
_ ≤ ∫⁻ X, ∫⁻ z : Label k, (1 : ℝ≥0∞) ∂unitBallLaw k
∂Measure.pi (fun _ : Fin k => stdGaussian E) :=
lintegral_mono (fun X => lintegral_mono (fun z => hKsum _))
_ = 1 := by simp
theorem positive_family_cap_domination :
∃ C : ℝ, 0 < C ∧ ∀ d : ℕ, 32 ≤ d →
let k := d/16
let α : ℝ := ((d:ℝ)-1)/2
∀ (c : EuclideanSpace ℝ (Fin d)), ‖c‖ = 1 →
∀ r : ℝ, 0 < r → r ≤ 2 → ∀ N : ℕ,
∀ K : Fin N → (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞,
(∀ j, Measurable (K j)) → (∀ z, ∑ j, K j z ≤ 1) →
(∑ j, capReference (K j) c r) ≤ 1 ∧
∀ j, ∃ D : CapDictionary c r α, ∃ a : D.I → ℝ,
(∀ i, 0 ≤ a i) ∧ Summable (fun i => a i*(r/D.radius i)^α) ∧
(∑' i, a i*(r/D.radius i)^α) ≤ Real.exp (C*d)*(capReference (K j) c r).toReal^(1-1/(k:ℝ)) ∧
(capMeasure c r).withDensity (capKernelValue (K j)) ≤
Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i)) ∧
(Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i))) Set.univ < ∞ := by
obtain ⟨C,hC,h⟩ := positive_cap_domination
refine ⟨C,hC,?_⟩
intro d hd k α c hc r hr hr2 N K hK hKsum
refine ⟨capReference_sum_le_one K hK hKsum c r,?_⟩
intro j
apply h d hd c hc r hr hr2 (K j) (hK j)
intro z
exact (Finset.single_le_sum (fun i _ => bot_le) (Finset.mem_univ j)).trans (hKsum z)
end ProjectionMoments
end
end OAI
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