| import Mathlib | |
| import OAI.Probability.SKGap.Matrix.LogDetUniform | |
| import OAI.Probability.SKGap.Stability.FieldIntegralReduction | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace SKGap | |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal | |
| theorem field_logdet_uniform {j ε : ℝ} (hj : 0 < j) (hj1 : j < 1) (hε : 0 < ε) : | |
| ∃ γ : ℝ,0 < γ ∧ ∃ N : ℕ,0 < N ∧ ∀ n : ℕ,N ≤ n → | |
| (gaussianCoordinates (MatrixCoordinates (Fin n))).real | |
| {g | ∃ y : Field n,(n:ℝ)*(j*varianceAverage y^2+ε) < | |
| log ((fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y)ᵀ* | |
| fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y+γ • 1).det} ≤ | |
| 4*exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by | |
| obtain ⟨γ,hγ,hγbound⟩ := actual_simultaneous_logdet hj hj1 hε | |
| obtain ⟨N,hN,hbound⟩ := hγbound γ hγ le_rfl 2 (3*j) | |
| refine ⟨γ,hγ,N,hN,?_⟩ | |
| intro n hn | |
| have hn0 : 0 < n := hN.trans_le hn | |
| have hnreal : (0:ℝ) < n := Nat.cast_pos.mpr hn0 | |
| apply (measureReal_mono (s₂ := {g | ∃ a : Fin n → ℝ,(∀ i,0 ≤ a i) ∧ (∀ i,a i ≤ 1) ∧ | |
| ∃ E : Matrix (Fin n) (Fin n) ℝ,E.rank ≤ 2 ∧ opNorm E ≤ 3*j ∧ | |
| j*(coordAverage a)^2+ε < LogDet.regularizedLogDet γ | |
| (1+((j*coordAverage a) • 1-goeMatrix (j/(n:ℝ)) g)*diagonal a+E)}) ?_ | |
| (measure_ne_top _ _)).trans (hbound n hn) | |
| rintro g ⟨y,hy⟩ | |
| refine ⟨spinVariance y,fun i=>(spinVariance_pos y i).le,spinVariance_le_one y, | |
| plantedJacobianPerturbation j y,plantedJacobianPerturbation_rank j y, | |
| plantedJacobianPerturbation_norm hn0 hj.le y,?_⟩ | |
| rw [← fieldJacobian_planted_formula hn0] | |
| have hav : coordAverage (spinVariance y)=varianceAverage y := by | |
| simpa only [coordAverage,Fintype.card_fin] using (varianceAverage_eq_average hn0 y).symm | |
| rw [hav] | |
| simpa only [LogDet.regularizedLogDet,LogDet.regularizedGram,Fintype.card_fin,lt_div_iff₀ hnreal,mul_comm] using hy | |
| end SKGap | |
| end | |
| end | |
| section | |
| noncomputable section | |
| namespace SKGap | |
| open Matrix Real Set | |
| open scoped BigOperators | |
| lemma spinVariance_difference_norm {n : ℕ} (y d : Field n) : | |
| vectorNorm (spinVariance (y+d)-spinVariance y) ≤ 2*vectorNorm d := by | |
| apply vectorNorm_le_of_sq_sum (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (vectorNorm_nonneg _)) | |
| rw [mul_pow,vectorNorm_sq,vectorSqNorm,Finset.mul_sum] | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| have hh : |spinVariance (y+d) i-spinVariance y i| ≤ 2*|d i| := by | |
| simpa only [spinVariance,magnetization,Pi.add_apply,add_sub_cancel_left] using | |
| scalar_variance_lipschitz_abs (y i+d i) (y i) | |
| have ha := mul_self_le_mul_self (abs_nonneg _) hh | |
| simpa only [sq_abs,← pow_two,mul_pow,Pi.sub_apply] using ha | |
| lemma dot_magnetization_sq_difference {n : ℕ} (y d v : Field n) : | |
| |((∑ i,magnetization (y+d) i*v i)^2)-(∑ i,magnetization y i*v i)^2| ≤ | |
| 2*sqrt (n:ℝ)*vectorNorm d*vectorSqNorm v := by | |
| let a := ∑ i,magnetization (y+d) i*v i | |
| let b := ∑ i,magnetization y i*v i | |
| have ha : |a| ≤ sqrt (n:ℝ)*vectorNorm v := | |
| (vector_dot_abs_le (magnetization (y+d)) v).trans | |
| (mul_le_mul_of_nonneg_right (magnetization_norm_bound (y+d)) (vectorNorm_nonneg _)) | |
| have hb : |b| ≤ sqrt (n:ℝ)*vectorNorm v := | |
| (vector_dot_abs_le (magnetization y) v).trans | |
| (mul_le_mul_of_nonneg_right (magnetization_norm_bound y) (vectorNorm_nonneg _)) | |
| have hab : |a-b| ≤ vectorNorm d*vectorNorm v := by | |
| have he : a-b=∑ i,(magnetization (y+d)-magnetization y) i*v i := by | |
| simp only [a,b,Pi.sub_apply,sub_mul,Finset.sum_sub_distrib] | |
| rw [he] | |
| exact (vector_dot_abs_le _ _).trans | |
| (mul_le_mul_of_nonneg_right (magnetization_difference_norm y d) (vectorNorm_nonneg _)) | |
| have hsum : |a+b| ≤ 2*sqrt (n:ℝ)*vectorNorm v := (abs_add_le _ _).trans (by linarith) | |
| change |a^2-b^2| ≤ _ | |
| rw [sq_sub_sq,abs_mul] | |
| apply (mul_le_mul hsum hab (abs_nonneg _) (mul_nonneg (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (sqrt_nonneg _)) (vectorNorm_nonneg _))).trans_eq | |
| rw [← vectorNorm_sq] | |
| ring | |
| lemma hessian_field_difference {n : ℕ} (hn : 0 < n) {j A : ℝ} (hj : 0 ≤ j) (_hA : 0 ≤ A) | |
| (J : Matrix (Fin n) (Fin n) ℝ) (y d a x : Field n) | |
| (ha0 : ∀ i,0 ≤ a i) (haA : ∀ i,a i ≤ A) : | |
| |quadraticForm (fieldHessian j J (y+d) a) x-quadraticForm (fieldHessian j J y a) x| ≤ | |
| (6*j*A/sqrt (n:ℝ))*vectorNorm d*vectorSqNorm x := by | |
| let v : Field n := fun i=>sqrt (a i)*x i | |
| have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr hn | |
| have hs0 : 0 < sqrt (n:ℝ) := sqrt_pos.mpr hn0 | |
| have hv : vectorSqNorm v ≤ A*vectorSqNorm x := by | |
| unfold vectorSqNorm v | |
| rw [Finset.mul_sum] | |
| exact Finset.sum_le_sum (fun i _=>by | |
| rw [mul_pow,sq_sqrt (ha0 i)] | |
| exact mul_le_mul_of_nonneg_right (haA i) (sq_nonneg _)) | |
| have hb : |onsager j (y+d)-onsager j y| ≤ j*(2/sqrt (n:ℝ)*vectorNorm d) := by | |
| unfold onsager | |
| rw [← mul_sub,abs_mul,abs_of_nonneg hj] | |
| exact mul_le_mul_of_nonneg_left (varianceAverage_difference hn y d) hj | |
| have he : quadraticForm (fieldHessian j J (y+d) a) x-quadraticForm (fieldHessian j J y a) x= | |
| (onsager j (y+d)-onsager j y)*vectorSqNorm v- | |
| (2*j/(n:ℝ))*((∑ i,magnetization (y+d) i*v i)^2-(∑ i,magnetization y i*v i)^2) := by | |
| rw [hessian_quadratic_formula,hessian_quadratic_formula] | |
| dsimp [v] | |
| ring | |
| rw [he] | |
| apply (abs_sub _ _).trans | |
| rw [abs_mul,abs_of_nonneg (vectorSqNorm_nonneg v),abs_mul, | |
| abs_of_nonneg (div_nonneg (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) hj) hn0.le)] | |
| have hq := dot_magnetization_sq_difference y d v | |
| have hfirst := mul_le_mul_of_nonneg_right hb (vectorSqNorm_nonneg v) | |
| have hsecond := mul_le_mul_of_nonneg_left hq (div_nonneg (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) hj) hn0.le) | |
| have hfactor : j*(2/sqrt (n:ℝ)*vectorNorm d)*vectorSqNorm v+ | |
| (2*j/(n:ℝ))*(2*sqrt (n:ℝ)*vectorNorm d*vectorSqNorm v)= | |
| (6*j/sqrt (n:ℝ))*vectorNorm d*vectorSqNorm v := by | |
| field_simp | |
| rw [sq_sqrt hn0.le] | |
| ring | |
| apply (add_le_add hfirst hsecond).trans | |
| rw [hfactor] | |
| calc | |
| _ ≤ (6*j/sqrt (n:ℝ))*vectorNorm d*(A*vectorSqNorm x) := | |
| mul_le_mul_of_nonneg_left hv (mul_nonneg (by positivity) (vectorNorm_nonneg _)) | |
| _ = _ := by ring | |
| lemma badField_neighborhood {n : ℕ} (hn : 0 < n) {j A ε c r : ℝ} | |
| (hj : 0 ≤ j) (hA : 0 ≤ A) (hε : 0 < ε) (_hr0 : 0 ≤ r) | |
| (hrε : 2*r ≤ ε/2) (hrc : 6*j*A*r ≤ c/2) | |
| (W : Matrix (Fin n) (Fin n) ℝ) (y d : Field n) | |
| (hy : (W,y)∈badFieldSet n j A (ε/2) (c/2)) | |
| (hd : vectorNorm d ≤ r*sqrt (n:ℝ)) : | |
| (W,y+d)∈badFieldSet n j A ε c := by | |
| obtain ⟨a,ha,hclose,x,hx,hquad⟩ := hy | |
| have hs := sqrt_nonneg (n:ℝ) | |
| have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr hn | |
| have hnorm : vectorNorm (a-spinVariance y) ≤ (ε/2)*sqrt (n:ℝ) := by | |
| apply vectorNorm_le_of_sq_sum (by positivity) | |
| simpa only [Pi.sub_apply,mul_pow,sq_sqrt hn0.le] using hclose | |
| have hdist : vectorNorm (a-spinVariance (y+d)) ≤ ε*sqrt (n:ℝ) := by | |
| have he : a-spinVariance (y+d)=(a-spinVariance y)-(spinVariance (y+d)-spinVariance y) := by abel | |
| rw [he] | |
| apply (vectorNorm_sub_le _ _).trans | |
| have hm := (spinVariance_difference_norm y d).trans (mul_le_mul_of_nonneg_left hd (by norm_num)) | |
| nlinarith [mul_le_mul_of_nonneg_right hrε hs] | |
| refine ⟨a,ha,?_,x,hx,?_⟩ | |
| · have hh := mul_self_le_mul_self (vectorNorm_nonneg (a-spinVariance (y+d))) hdist | |
| simpa only [← pow_two,vectorNorm_sq,vectorSqNorm,Pi.sub_apply,mul_pow,sq_sqrt hn0.le] using hh | |
| · have hv : vectorSqNorm x.ofLp=1 := by | |
| rw [← vectorNorm_sq] | |
| change ‖WithLp.toLp 2 x.ofLp‖^2=1 | |
| simp only [WithLp.toLp_ofLp,hx,one_pow] | |
| have hb := hessian_field_difference hn hj hA (plantedInteraction j W) y d a x.ofLp | |
| (fun i=>(ha i).1) (fun i=>(ha i).2) | |
| rw [hv,mul_one] at hb | |
| have hc := (mul_le_mul_of_nonneg_left hd (by positivity : 0 ≤ 6*j*A/sqrt (n:ℝ))) | |
| have he : (6*j*A/sqrt (n:ℝ))*(r*sqrt (n:ℝ))=6*j*A*r := by field_simp | |
| rw [he] at hc | |
| linarith [le_abs_self (quadraticForm (fieldHessian j (plantedInteraction j W) (y+d) a) x.ofLp- | |
| quadraticForm (fieldHessian j (plantedInteraction j W) y a) x.ofLp)] | |
| end SKGap | |
| end | |
| end | |
| end OAI | |