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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 | import Mathlib
import OAI.Probability.SKGap.Localization.PlantedFixedTime
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real Set
open scoped BigOperators
def rootBad {n : ℕ} (j A ε c ρ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) (h : Field n) : Prop :=
∃ y : Field n,vectorNorm (tapField j J h y) ≤ ρ*sqrt (n:ℝ) ∧
∃ a : Field n,(∀ i,a i∈Icc 0 A) ∧
(∑ i,(a i-spinVariance y i)^2) ≤ ε^2*(n:ℝ) ∧
∃ x : EuclideanSpace ℝ (Fin n),‖x‖=1 ∧ quadraticForm (fieldHessian j J y a) x.ofLp ≤ c
lemma rootBad_diagonal_transfer {n : ℕ} {j A ε c ρ δ : ℝ} (hA : 0 ≤ A) (hδ : 0 ≤ δ)
(J : Matrix (Fin n) (Fin n) ℝ) (h d : Field n) (hd : ∀ i,|d i| ≤ δ) :
rootBad j A ε c ρ J h → rootBad j A ε (c+A*δ) (ρ+δ) (J+diagonal d) h := by
rintro ⟨y,hy,a,ha,hdist,x,hx,hq⟩
refine ⟨y,?_,a,ha,hdist,x,hx,?_⟩
· have hdiff := tapField_diagonal_difference_bound hδ j J h y d hd
have hh := vectorNorm_add_le (tapField j J h y)
(tapField j (J+diagonal d) h y-tapField j J h y)
rw [add_sub_cancel] at hh
exact (hh.trans (add_le_add hy hdiff)).trans_eq (by ring)
· have hdiff := hessian_diagonal_difference_bound hA hδ j J y a d x.ofLp
(fun i=>(ha i).1) (fun i=>(ha i).2) hd
have hv : vectorSqNorm x.ofLp=1 := by
rw [← vectorNorm_sq]
change ‖x‖^2=1
rw [hx];norm_num
rw [hv,mul_one] at hdiff
linarith only [le_abs_self (quadraticForm (fieldHessian j (J+diagonal d) y a) x.ofLp-
quadraticForm (fieldHessian j J y a) x.ofLp),hdiff,hq]
lemma rootBad_mono {n : ℕ} {j A ε c c' ρ ρ' : ℝ}
(hc : c ≤ c') (hρ : ρ ≤ ρ') (J : Matrix (Fin n) (Fin n) ℝ) (h : Field n) :
rootBad j A ε c ρ J h → rootBad j A ε c' ρ' J h := by
rintro ⟨y,hy,a,ha,hd,x,hx,hq⟩
exact ⟨y,hy.trans (mul_le_mul_of_nonneg_right hρ (sqrt_nonneg _)),a,ha,hd,x,hx,hq.trans hc⟩
def eraseDiagonal {n : ℕ} (J : Matrix (Fin n) (Fin n) ℝ) : Matrix (Fin n) (Fin n) ℝ :=
J-diagonal (fun i=>J i i)
lemma eraseDiagonal_add {n : ℕ} (J : Matrix (Fin n) (Fin n) ℝ) :
eraseDiagonal J+diagonal (fun i=>J i i)=J := sub_add_cancel _ _
lemma eraseDiagonal_diag {n : ℕ} (J : Matrix (Fin n) (Fin n) ℝ) (i : Fin n) :
eraseDiagonal J i i=0 := by simp [eraseDiagonal]
lemma erased_planted_witness {n : ℕ} {j A K ε c ρ δ : ℝ}
(hA : 0 ≤ A) (hδ : 0 ≤ δ) (hδρ : δ ≤ ρ/2) (hδc : A*δ ≤ c/2)
(t : ℝ) (g : MatrixCoordinates (Fin n)→ℝ) (b : Field n)
(hd : ∀ i,|plantedInteraction j (goeMatrix (j/(n:ℝ)) g) i i| ≤ δ)
(hJ : ∀ z : Field n,vectorNorm ((eraseDiagonal (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)))*ᵥz) ≤
(K-δ)*vectorNorm z)
(hbad : rootBad j A ε (c/2) (ρ/2)
(eraseDiagonal (plantedInteraction j (goeMatrix (j/(n:ℝ)) g))) (fun i=>t+b i)) :
(g,b)∈plantedRootBad n j A K ε c ρ t := by
let J := plantedInteraction j (goeMatrix (j/(n:ℝ)) g)
have hjnorm := mulvec_add_diagonal_bound hδ (eraseDiagonal J) (fun i=>J i i) hJ hd
rw [eraseDiagonal_add] at hjnorm
have hjnorm' : ∀ z : Field n, vectorNorm (J*ᵥz) ≤ K*vectorNorm z := by
simpa only [sub_add_cancel] using hjnorm
have hh := rootBad_diagonal_transfer hA hδ (eraseDiagonal J) (fun i=>t+b i) (fun i=>J i i) hd hbad
rw [eraseDiagonal_add] at hh
obtain ⟨y,hy,a,ha,hdist,x,hx,hq⟩ := rootBad_mono (by linarith : c/2+A*δ ≤ c)
(by linarith : ρ/2+δ ≤ ρ) J (fun i=>t+b i) hh
exact ⟨hjnorm',y,hy,a,ha,hdist,x,hx,hq⟩
end SKGap
end
end
end OAI
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