| import Mathlib | |
| import OAI.Probability.SKGap.Localization.ConditionalHessianLaw | |
| import OAI.Probability.SKGap.Localization.ScalarEmpirical | |
| import OAI.Probability.SKGap.Gaussian.ScalarGaussianReductionAlgebra | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace SKGap | |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal | |
| lemma tapField_planted_observation {n : ℕ} [NeZero n] (j t σ : ℝ) | |
| (y x : Fin n → ℝ) (W : Matrix (Fin n) (Fin n) ℝ) : | |
| tapField j (plantedInteraction j W) (fun i=>t+x i) y= | |
| empiricalObservation j t σ y-W*ᵥmagnetization y-x := by | |
| ext i | |
| simp only [tapField,plantedInteraction,Matrix.add_apply,Matrix.smul_apply, | |
| smul_eq_mul,vecMulVec_apply,mul_one,add_mul,Finset.sum_add_distrib, | |
| ← Finset.mul_sum,empiricalObservation,scalarD,scalarM,integral_empiricalLaw, | |
| Fintype.card_fin,scalarBMoment,scalarQMoment_empirical_overlap,onsager, | |
| magnetization,Pi.sub_apply,Matrix.mulVec,dotProduct] | |
| ring | |
| lemma scalar_density_cancellation {n : ℕ} [NeZero n] {j t σ : ℝ} | |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (y : Fin n → ℝ) : | |
| rankOneDensity ((j/(n:ℝ))*(magnetization y⬝ᵥmagnetization y)+(t+σ^2)) | |
| (j/(n:ℝ)) (magnetization y) (empiricalObservation j t σ y)* | |
| exp ((n:ℝ)*j*(1-overlap y)^2/2)= | |
| sqrt (scalarS j (scalarEmpirical y t σ)/ | |
| (scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y)))*scalarIntegrand j t σ y := by | |
| let p := scalarEmpirical y t σ | |
| let q := scalarQMoment (empiricalLaw y) | |
| let s := scalarS j p | |
| let S := s+j*q | |
| have hn : (n:ℝ) ≠ 0 := by exact_mod_cast NeZero.ne n | |
| have hq : 0 ≤ q := (scalarQMoment_bounds _).1 | |
| have hs : 0 < s := add_pos_of_pos_of_nonneg (add_pos_of_nonneg_of_pos ht (sq_pos_of_pos hσ)) (mul_nonneg hj hq) | |
| have hS : 0 < S := add_pos_of_pos_of_nonneg hs (mul_nonneg hj hq) | |
| have hm : magnetization y⬝ᵥmagnetization y=(n:ℝ)*q := by | |
| dsimp [q] | |
| rw [scalarQMoment_empirical_overlap] | |
| simp only [overlap,magnetization,dotProduct,← pow_two] | |
| field_simp | |
| have hx : (fun i=>y i-scalarD j p)⬝ᵥmagnetization y=(n:ℝ)*scalarA j p := by | |
| rw [dotProduct_comm] | |
| exact scalarA_empirical_sum j t σ y | |
| have hs' : (j/(n:ℝ))*(magnetization y⬝ᵥmagnetization y)+(t+σ^2)=s := by | |
| rw [hm]; dsimp [s,scalarS,p,scalarEmpirical,q]; field_simp; ring | |
| rw [hs'] | |
| have hobs : empiricalObservation j t σ y=(fun i=>y i-scalarD j p)+(j*(1-q)) • magnetization y := rfl | |
| rw [hobs] | |
| have hc := rankOneDensity_cancellation hn hs hS.ne' (rfl : S=s+j*q) | |
| (fun i=>y i-scalarD j p) (magnetization y) hm hx (b := 1-q) | |
| have hb : 1-overlap y=1-q := by rw [← scalarQMoment_empirical_overlap] | |
| rw [hb,hc] | |
| have hd : productNormal (scalarD j p) s y=radialDensity (sqrt s) (fun i=>y i-scalarD j p) := by | |
| unfold productNormal | |
| rw [show s.toNNReal=NNReal.mk s hs.le from by ext; exact Real.coe_toNNReal _ hs.le] | |
| exact gaussian_density_shift _ hs.le y | |
| rw [scalarIntegrand,show scalarEmpirical y t σ=p from rfl,hd] | |
| simp only [Fintype.card_fin] | |
| have he : (n:ℝ)*j*(scalarA j p-s*(1-q))^2/(2*s*S)=(n:ℝ)*scalarPsi j p := by | |
| dsimp [S,s,q,p,scalarPsi,scalarEmpirical] | |
| ring | |
| rw [he] | |
| ring | |
| end SKGap | |
| end | |
| end | |
| end OAI | |