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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Hierarchy.WeightedMoments
import OAI.Probability.SKBarriers.Coverage.PaleyZygmund
import OAI.Probability.SKBarriers.Coverage.WeightedConcentration
import OAI.Probability.SKBarriers.Coverage.SubGaussianQuantile
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem weightedPartition_pos {n : ℕ} (ν : Config n → ℝ)
(hνpos : ∀ x, 0 ≤ ν x) (hν : ∑ x, ν x = 1) (b : ℝ) (G : Disorder n) :
0 < weightedPartition ν b G := by
let : Weighted.Weights n := ⟨ν,hνpos,hν⟩
exact Weighted.partition_pos b G
theorem continuous_weightedPartition {n : ℕ} (ν : Config n → ℝ) (b : ℝ) :
Continuous (weightedPartition ν b) := by
unfold weightedPartition
exact continuous_finsetSum _ (fun x _ => continuous_const.mul
(Real.continuous_exp.comp (continuous_const.mul (continuous_hamiltonian x))))
theorem weightedLogPartition_subGaussian {n : ℕ} (hn : 0 < n)
(ν : Config n → ℝ) (hνpos : ∀ x, 0 ≤ ν x) (hν : ∑ x, ν x = 1) (b : ℝ) :
HasSubgaussianMGF (fun G : Disorder n => Real.log (weightedPartition ν b G)-
∫ K, Real.log (weightedPartition ν b K) ∂disorderLaw n)
(logPartitionProxy b n) (disorderLaw n) := by
let : Weighted.Weights n := ⟨ν,hνpos,hν⟩
exact Weighted.logPartition_subGaussian hn b
theorem weightedPartition_paleyZygmund {n : ℕ} (hn : 0 < n)
(ν : Config n → ℝ) (hνpos : ∀ x, 0 ≤ ν x) (hν : ∑ x, ν x = 1)
(b q : ℝ) (hq : 0 ≤ q)
(hr : weightedOverlapMass ν q*Real.exp (b^2*(n:ℝ)/2) ≤
Real.exp (b^2*(n:ℝ)*q^2/2)) :
Real.exp (-(b^2*(n:ℝ)*q^2/2+Real.log 8)) ≤
(disorderLaw n).real {G | b^2*((n:ℝ)-1)/4-Real.log 2 ≤
Real.log (weightedPartition ν b G)} := by
let M := Real.exp (b^2*((n:ℝ)-1)/4)
let E := Real.exp (b^2*(n:ℝ)*q^2/2)
let S := ∫ G, (weightedPartition ν b G)^2 ∂disorderLaw n
have hM : 0 < M := Real.exp_pos _
have hE : 0 < E := Real.exp_pos _
have hS : 0 < S := by
apply (integral_pos_iff_support_of_nonneg (fun G => sq_nonneg _)
(weightedPartition_sq_integrable ν b)).mpr
have hsupp : Function.support (fun G => (weightedPartition ν b G)^2) = Set.univ := by
ext G
simp only [Function.mem_support,Set.mem_univ,iff_true]
exact pow_ne_zero 2 (weightedPartition_pos ν hνpos hν b G).ne'
rw [hsupp]
simp
have hm := weightedPartition_mean hn ν hν b
have hR := (weightedPartition_secondMoment_le hn ν hνpos hν b q hq).trans
(add_le_add_right hr _)
rw [hm] at hR
have hR' : S ≤ (2*E)*M^2 := by
apply (div_le_iff₀ (sq_pos_of_pos hM)).mp
change S/M^2 ≤ 2*E
change S/M^2 ≤ E+E at hR
linarith
have hp := paleyZygmund_half (disorderLaw n) (weightedPartition ν b)
(continuous_weightedPartition ν b).measurable
(fun G => (weightedPartition_pos ν hνpos hν b G).le)
(weightedPartition_integrable ν b) (weightedPartition_sq_integrable ν b)
(by rw [hm]; exact hM)
rw [hm] at hp
have he : Real.exp (-(b^2*(n:ℝ)*q^2/2+Real.log 8)) = 1/(8*E) := by
rw [Real.exp_neg,Real.exp_add,Real.exp_log (by norm_num)]
dsimp only [E]
ring
have hbound : Real.exp (-(b^2*(n:ℝ)*q^2/2+Real.log 8)) ≤ M^2/(4*S) := by
rw [he]
apply (div_le_div_iff₀ (by positivity : 0 < 8*E) (by positivity : 0 < 4*S)).mpr
nlinarith
have hex : Real.exp (b^2*((n:ℝ)-1)/4-Real.log 2) = M/2 := by
rw [Real.exp_sub,Real.exp_log (by norm_num)]
have hevent : {G : Disorder n | M/2 ≤ weightedPartition ν b G} =
{G | b^2*((n:ℝ)-1)/4-Real.log 2 ≤ Real.log (weightedPartition ν b G)} := by
ext G
simp only [Set.mem_ofPred_eq]
symm
rw [← Real.exp_le_exp,
Real.exp_log (weightedPartition_pos ν hνpos hν b G),hex]
rw [← hevent]
exact hbound.trans hp
theorem weightedPartition_half_gain {n : ℕ} (hn : 0 < n)
(ν : Config n → ℝ) (hνpos : ∀ x, 0 ≤ ν x) (hν : ∑ x, ν x = 1)
(b q : ℝ) (hb : b ≠ 0) (hq : 0 ≤ q)
(hr : weightedOverlapMass ν q*Real.exp (b^2*(n:ℝ)/2) ≤
Real.exp (b^2*(n:ℝ)*q^2/2)) :
(1/2:ℝ) ≤ (disorderLaw n).real {G | b^2*((n:ℝ)-1)/4-Real.log 2-
Real.sqrt (2*(logPartitionProxy b n:ℝ)*(b^2*(n:ℝ)*q^2/2+Real.log 8))-
Real.sqrt (2*(logPartitionProxy b n:ℝ)*Real.log 2) ≤
Real.log (weightedPartition ν b G)} := by
apply subGaussian_half_lower
((continuous_weightedPartition ν b).log
(fun G => (weightedPartition_pos ν hνpos hν b G).ne')).measurable
(show 0 < (logPartitionProxy b n:ℝ) from by
change 0 < Real.pi^2*b^2*(n:ℝ)/8
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hb' := sq_pos_of_ne_zero hb
positivity)
(weightedLogPartition_subGaussian hn ν hνpos hν b)
(by positivity)
(weightedPartition_paleyZygmund hn ν hνpos hν b q hq hr)
end SK.Analytic
end
end
end
end OAI
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