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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Stability.RootDiagonalTransfer
import OAI.Probability.SKGap.Matrix.GOESubmatrix
namespace OAI
section
noncomputable section
namespace SKGap.ObservationBridge
open Matrix Real Set MeasureTheory ProbabilityTheory Filter
open scoped BigOperators ENNReal Topology
lemma absorb_linear_exponential {b : ℝ} (hb : 0 < b) :
∃ N : ℕ, ∀ n ≥ N,
2*(n:ℝ)*exp (-b*(n:ℝ)) ≤ exp (-(b/2)*(n:ℝ)) := by
obtain ⟨N,hN⟩ := exists_nat_ge (16/b^2)
refine ⟨N,fun n hn=>?_⟩
have hnR : 16/b^2 ≤ (n:ℝ) := hN.trans (Nat.cast_le.mpr hn)
have hn0 : 0 ≤ (n:ℝ) := Nat.cast_nonneg _
have hbn : 16 ≤ (n:ℝ)*b^2 := (div_le_iff₀ (sq_pos_of_pos hb)).mp hnR
have hmul := mul_le_mul_of_nonneg_right hbn hn0
have he := Real.pow_div_factorial_le_exp ((b/2)*(n:ℝ)) (by positivity) 2
norm_num only [Nat.factorial_succ,Nat.factorial_zero,Nat.cast_mul,Nat.cast_one,Nat.cast_ofNat] at he
have hhead : 2*(n:ℝ) ≤ exp ((b/2)*(n:ℝ)) := by nlinarith
calc
_ ≤ exp ((b/2)*(n:ℝ))*exp (-b*(n:ℝ)) :=
mul_le_mul_of_nonneg_right hhead (exp_pos _).le
_ = _ := by rw [← exp_add]; congr 1; ring
lemma shifted_goe_diagonal_tail {j δ : ℝ} (hj : 0 < j) (hδ : 0 < δ)
{n : ℕ} (hn : 0 < n) (hmean : j/(n:ℝ) ≤ δ/2) :
(gaussianCoordinates (MatrixCoordinates (Fin n))).real
{g | ∃ i, δ < |plantedInteraction j (goeMatrix (j/(n:ℝ)) g) i i|} ≤
2*(n:ℝ)*exp (-(δ^2/(32*j))*(n:ℝ)) := by
have hr : 0 < j/(n:ℝ) := div_pos hj (Nat.cast_pos.mpr hn)
have hs : {g : MatrixCoordinates (Fin n)→ℝ | ∃ i,
δ < |plantedInteraction j (goeMatrix (j/(n:ℝ)) g) i i|} ⊆
{g | ∃ i, δ/2 < |goeMatrix (j/(n:ℝ)) g i i|} := by
rintro g ⟨i,hi⟩
refine ⟨i,?_⟩
have he : plantedInteraction j (goeMatrix (j/(n:ℝ)) g) i i =
goeMatrix (j/(n:ℝ)) g i i+j/(n:ℝ) := by
simp [plantedInteraction,Matrix.vecMulVec]
rw [he] at hi
have hh := abs_add_le (goeMatrix (j/(n:ℝ)) g i i) (j/(n:ℝ))
rw [abs_of_pos hr] at hh
linarith
apply (measureReal_mono hs (measure_ne_top _ _)).trans
have hh := goe_diagonal_tail (ι:=Fin n) hr (half_pos hδ).le
have he : -(δ/2)^2/(8*(j/(n:ℝ))) = -(δ^2/(32*j))*(n:ℝ) := by
field_simp
ring
simpa only [Fintype.card_fin,he] using hh
def erasedPlantedRootBad (n : ℕ) (j A K ε c ρ t : ℝ) :
Set ((MatrixCoordinates (Fin n)→ℝ) × (Fin n→ℝ)) :=
{p | (∀ z : Field n, vectorNorm
(eraseDiagonal (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))*ᵥz) ≤ K*vectorNorm z) ∧
rootBad j A ε c ρ (eraseDiagonal (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)))
(fun i=>t+p.2 i)}
lemma erased_planted_probability {n : ℕ} {j A K ε c ρ δ a t : ℝ}
(hj : 0 < j) (hA : 0 ≤ A) (hδ : 0 < δ) (hn : 0 < n)
(hδρ : δ ≤ ρ/2) (hδc : A*δ ≤ c/2) (hmean : j/(n:ℝ) ≤ δ/2)
(hp : ((gaussianCoordinates (MatrixCoordinates (Fin n))).prod
(Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real
(plantedRootBad n j A K ε c ρ t) ≤ exp (-a*(n:ℝ))) :
((gaussianCoordinates (MatrixCoordinates (Fin n))).prod
(Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real
(erasedPlantedRootBad n j A (K-δ) ε (c/2) (ρ/2) t) ≤
exp (-a*(n:ℝ))+2*(n:ℝ)*exp (-(δ^2/(32*j))*(n:ℝ)) := by
let μ := gaussianCoordinates (MatrixCoordinates (Fin n))
let ν := Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal)
let D := {g : MatrixCoordinates (Fin n)→ℝ | ∃ i,
δ < |plantedInteraction j (goeMatrix (j/(n:ℝ)) g) i i|}
have hs : erasedPlantedRootBad n j A (K-δ) ε (c/2) (ρ/2) t ⊆
plantedRootBad n j A K ε c ρ t ∪ (D ×ˢ (Set.univ : Set (Fin n→ℝ))) := by
intro p hp
by_cases hd : p.1∈D
· exact Or.inr ⟨hd,mem_univ _⟩
· apply Or.inl
exact erased_planted_witness hA hδ.le hδρ hδc t p.1 p.2
(fun i=>le_of_not_gt (fun hi=>hd ⟨i,hi⟩)) hp.1 hp.2
have hD : (μ.prod ν).real (D ×ˢ (Set.univ : Set (Fin n→ℝ))) ≤
2*(n:ℝ)*exp (-(δ^2/(32*j))*(n:ℝ)) := by
rw [measureReal_prod_prod]
simp only [Measure.real,measure_univ,ENNReal.toReal_one,mul_one]
exact shifted_goe_diagonal_tail hj hδ hn hmean
exact (measureReal_mono hs (measure_ne_top _ _)).trans
((measureReal_union_le _ _).trans (add_le_add hp hD))
theorem zero_diagonal_fixed_time_root_stability {j T : ℝ}
(hj : 0 < j) (hj1 : j < 1) (hT : 0 < T) :
∃ A K ε c ρ a : ℝ, 1 < A ∧ 2*sqrt j < K ∧ sqrt j*A < 1 ∧
0 < ε ∧ 0 < c ∧ 0 < ρ ∧ 0 < a ∧ ∃ N : ℕ, 0 < N ∧
∀ n ≥ N, ∀ t∈Icc 0 T,
((gaussianCoordinates (MatrixCoordinates (Fin n))).prod
(Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real
(erasedPlantedRootBad n j A K ε c ρ t) ≤ exp (-a*(n:ℝ)) := by
obtain ⟨A,K,ε,c,ρ,a,hA,hK,hsub,hε,hc,hρ,ha,N,hN,hp⟩ :=
planted_fixed_time_root_stability hj hj1 hT
let δ := min ((K-2*sqrt j)/2) (min (ρ/2) (c/(2*A)))
have hA0 : 0 < A := zero_lt_one.trans hA
have hδ : 0 < δ := lt_min (by linarith) (lt_min (half_pos hρ) (by positivity))
have hδK : δ ≤ (K-2*sqrt j)/2 := min_le_left _ _
have hδρ : δ ≤ ρ/2 := (min_le_right _ _).trans (min_le_left _ _)
have hδc' : δ ≤ c/(2*A) := (min_le_right _ _).trans (min_le_right _ _)
have hδc : A*δ ≤ c/2 := by
have hh := (le_div_iff₀ (by positivity : 0 < 2*A)).mp hδc'
nlinarith
let b := δ^2/(32*j)
have hb : 0 < b := by dsimp [b]; positivity
obtain ⟨Nd,hNd⟩ := absorb_linear_exponential hb
obtain ⟨Nm,hNm⟩ := exists_nat_ge (2*j/δ)
obtain ⟨r,hr,Nr,hNr⟩ := two_exponential_tails ha (half_pos hb)
refine ⟨A,K-δ,ε,c/2,ρ/2,r,hA,by linarith,hsub,hε,half_pos hc,half_pos hρ,hr,
max N (max Nd (max Nm Nr)),hN.trans_le (le_max_left _ _),?_⟩
intro n hn t ht
have hnN : N ≤ n := (le_max_left _ _).trans hn
have hnD : Nd ≤ n := (le_max_left _ _).trans ((le_max_right _ _).trans hn)
have hnM : Nm ≤ n := (le_max_left _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hn))
have hnR : Nr ≤ n := (le_max_right _ _).trans
((le_max_right _ _).trans ((le_max_right _ _).trans hn))
have hn0 : 0 < n := hN.trans_le hnN
have hmean : j/(n:ℝ) ≤ δ/2 := by
have hh := (div_le_iff₀ hδ).mp (hNm.trans (Nat.cast_le.mpr hnM))
apply (div_le_iff₀ (Nat.cast_pos.mpr hn0)).mpr
nlinarith
have hh := erased_planted_probability hj hA0.le hδ hn0 hδρ hδc hmean (hp n hnN t ht)
apply hh.trans
apply (add_le_add_right (hNd n hnD) _).trans
exact (by nlinarith only [exp_pos (-a*(n:ℝ))] :
exp (-a*(n:ℝ))+exp (-(b/2)*(n:ℝ)) ≤
4*exp (-a*(n:ℝ))+exp (-(b/2)*(n:ℝ))).trans (hNr n hnR)
end SKGap.ObservationBridge
end
end
end OAI
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