| import Mathlib |
| import OAI.Probability.SKGap.Stability.RootDiagonalTransfer |
| import OAI.Probability.SKGap.Matrix.GOESubmatrix |
|
|
| namespace OAI |
|
|
| section |
|
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| 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 |
|
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| end |
| end |
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| end OAI |
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