| 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] |
| 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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