| import Mathlib |
| import OAI.Probability.SKRatio.Calculus.RowControl |
| import OAI.Probability.SKRatio.Variational.CompactField |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Set Real |
| namespace SKRatio.Planted |
| open Scalar Calculus |
| attribute [local instance] Classical.propDecidable |
| variable {n : ℕ} |
|
|
| def quadratic (A : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : ℝ := |
| ∑ i, ∑ j, A i j*p i*p j |
| |
| def weightedForm (A : Matrix (Fin n) (Fin n) ℝ) (G : FieldWeight × FieldWeight → ℝ) |
| (x : Fin n → FieldWeight) (p : Fin n → ℝ) : ℝ := |
| quadratic (fun i j => A i j*G (x i,x j)) p |
|
|
| def diagonalForm (A : Matrix (Fin n) (Fin n) ℝ) (G : FieldWeight × FieldWeight → ℝ) |
| (x : Fin n → FieldWeight) (p : Fin n → ℝ) : ℝ := |
| ∑ i, (∑ j, A i j*G (x i,x j))*p i^2 |
|
|
| def varianceKernel (p : FieldWeight × FieldWeight) : ℝ := compactV p.2 |
| def meanVarianceKernel (p : FieldWeight × FieldWeight) : ℝ := |
| (1-(p.2:ℝ))*compactV p.2 |
|
|
| lemma continuous_varianceKernel : Continuous varianceKernel := |
| continuous_compactV.comp continuous_snd |
| lemma continuous_meanVarianceKernel : Continuous meanVarianceKernel := by |
| unfold meanVarianceKernel |
| exact (continuous_const.sub (continuous_subtype_val.comp continuous_snd)).mul |
| (continuous_compactV.comp continuous_snd) |
|
|
| def compactForm (J : Interaction n) (x : Fin n → FieldWeight) (p : Fin n → ℝ) : ℝ := |
| weightedForm J varianceKernel x p + |
| 2*weightedForm (fun i j => J i j^2) meanVarianceKernel x p + |
| diagonalForm (fun i j => J i j^2) compactF x p + |
| weightedForm (fun i j => J i j^2) compactK x p |
|
|
| lemma quadratic_add (A B : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : |
| quadratic (A+B) p = quadratic A p+quadratic B p := by |
| simp only [quadratic,Matrix.add_apply,add_mul,Finset.sum_add_distrib] |
| lemma quadratic_sub (A B : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : |
| quadratic (A-B) p = quadratic A p-quadratic B p := by |
| simp only [quadratic,Matrix.sub_apply,sub_mul,Finset.sum_sub_distrib] |
| lemma quadratic_const_mul (a : ℝ) (A : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : |
| quadratic (a • A) p = a*quadratic A p := by |
| simp only [quadratic,Matrix.smul_apply,smul_eq_mul,mul_assoc,←Finset.mul_sum] |
| |
| lemma quadratic_inner (A : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : |
| quadratic A p = inner ℝ (WithLp.toLp 2 p) |
| (Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) A (WithLp.toLp 2 p)) := by |
| rw [Matrix.inner_toEuclideanCLM] |
| simp only [quadratic,dotProduct,Matrix.mulVec,Finset.mul_sum] |
| congr 1 |
| ext i |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
|
| lemma quadratic_abs_le (A : Matrix (Fin n) (Fin n) ℝ) (p : Fin n → ℝ) : |
| |quadratic A p| ≤ euclideanOpNorm A*(∑ i, p i^2) := by |
| rw [quadratic_inner] |
| let P : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 p |
| have hb := (abs_real_inner_le_norm P (Matrix.toEuclideanCLM (𝕜 := ℝ) A P)).trans |
| (mul_le_mul_of_nonneg_left ((Matrix.toEuclideanCLM (𝕜 := ℝ) A).le_opNorm P) (norm_nonneg P)) |
| calc |
| _ ≤ ‖P‖*(‖Matrix.toEuclideanCLM (𝕜 := ℝ) A‖*‖P‖) := hb |
| _ = euclideanOpNorm A*(∑ i, p i^2) := by |
| rw [show ‖P‖*(‖Matrix.toEuclideanCLM (𝕜 := ℝ) A‖*‖P‖) = |
| euclideanOpNorm A*‖P‖^2 by unfold euclideanOpNorm |
| rw [EuclideanSpace.real_norm_sq_eq] |
|
|
| lemma weightedForm_sub (A B : Matrix (Fin n) (Fin n) ℝ) |
| (G : FieldWeight × FieldWeight → ℝ) (x : Fin n → FieldWeight) (p : Fin n → ℝ) : |
| weightedForm A G x p-weightedForm B G x p = weightedForm (A-B) G x p := by |
| simp only [weightedForm,quadratic,Matrix.sub_apply,sub_mul,Finset.sum_sub_distrib] |
| |
| lemma diagonalForm_sub (A B : Matrix (Fin n) (Fin n) ℝ) |
| (G : FieldWeight × FieldWeight → ℝ) (x : Fin n → FieldWeight) (p : Fin n → ℝ) : |
| diagonalForm A G x p-diagonalForm B G x p = diagonalForm (A-B) G x p := by |
| simp only [diagonalForm,Matrix.sub_apply,sub_mul,Finset.sum_sub_distrib] |
| |
| lemma diagonalForm_abs_le (A : Matrix (Fin n) (Fin n) ℝ) |
| (G : FieldWeight × FieldWeight → ℝ) (x : Fin n → FieldWeight) (p : Fin n → ℝ) |
| {C : ℝ} (hC : ∀ i, |∑ j, A i j*G (x i,x j)| ≤ C) : |
| |diagonalForm A G x p| ≤ C*∑ i, p i^2 := by |
| apply (Finset.abs_sum_le_sum_abs _ _).trans |
| calc |
| _ ≤ ∑ i, C*p i^2 := Finset.sum_le_sum (fun i _ => by |
| rw [abs_mul,abs_of_nonneg (sq_nonneg (p i))] |
| exact mul_le_mul_of_nonneg_right (hC i) (sq_nonneg _)) |
| _ = _ := (Finset.mul_sum ..).symm |
|
|
| lemma sum_offdiag_eq_full {F : Fin n → Fin n → ℝ} (hF : ∀ i, F i i=0) : |
| (∑ i, ∑ j ∈ Finset.univ.erase i, F i j) = ∑ i, ∑ j, F i j := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| simpa only [hF i,add_zero] using Finset.sum_erase_add Finset.univ (F i) (Finset.mem_univ i) |
|
|
| lemma gradient_pair_expansion (J : Interaction n) (hJ : ∀ i j, J i j=J j i) |
| (hdiag : ∀ i, J i i=0) (H p : Fin n → ℝ) : |
| pairSum (fun i j => J i j^2*(v (H j)*p i+v (H i)*p j)^2/(w (H i)+w (H j))) = |
| diagonalForm (fun i j => J i j^2) compactF (fun i => compactWeight (H i)) p + |
| weightedForm (fun i j => J i j^2) compactK (fun i => compactWeight (H i)) p := by |
| have hd : ∀ i, J i i^2*(v (H i)^2/(w (H i)+w (H i)))*p i^2=0 := by simp [hdiag] |
| have hk : ∀ i, J i i^2*kernel (H i) (H i)*p i*p i=0 := by simp [hdiag] |
| simp only [diagonalForm,weightedForm,quadratic,compactF_weight,compactK_weight,Finset.sum_mul] |
| rw [←sum_offdiag_eq_full hd,←sum_offdiag_eq_full hk,sum_offdiag_pairs,sum_offdiag_pairs,←pairSum_add] |
| unfold pairSum |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| dsimp only [] |
| rw [hJ j i] |
| dsimp [kernel] |
| rw [add_comm (w (H j)) (w (H i))] |
| ring |
|
|
| theorem gradientForm_eq_compactForm (J : Interaction n) (hJ : ∀ i j, J i j=J j i) |
| (hdiag : ∀ i, J i i=0) (p : Fin n → ℝ) : |
| gradientForm J p = compactForm J (fun i => compactWeight (∑ j, J i j)) p := by |
| have hm (i : Fin n) : mean J (plusSpin n) i = m (∑ j, J i j) := by |
| simp [mean,field,plusSpin,spin,m] |
| have hv (i : Fin n) : conditionalVariance J (plusSpin n) i = v (∑ j, J i j) := by |
| simp only [conditionalVariance,hm,v] |
| have hw (i : Fin n) : gradientWeight J (plusSpin n) i = w (∑ j, J i j) := by |
| simp [gradientWeight,plusSpin,spin,hm,w] |
| simp only [gradientForm,hm,hv,hw,gradient_pair_expansion J hJ hdiag, |
| compactForm,weightedForm,quadratic,varianceKernel,meanVarianceKernel, |
| compactV_weight,compactWeight_coe] |
| have h₁ : (∑ i, ∑ j ∈ Finset.univ.erase i, J i j*p i*v (∑ k, J j k)*p j) = |
| ∑ i, ∑ j, J i j*v (∑ k, J j k)*p i*p j := by |
| rw [sum_offdiag_eq_full (by intro i |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| have h₂ : (∑ i, ∑ j ∈ Finset.univ.erase i, J i j^2*p i*m (∑ k, J j k)*v (∑ k, J j k)*p j) = |
| ∑ i, ∑ j, J i j^2*((1-w (∑ k, J j k))*v (∑ k, J j k))*p i*p j := by |
| rw [sum_offdiag_eq_full (by intro i |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [w] |
| ring |
| rw [h₁,h₂] |
| ring |
|
|
| end SKRatio.Planted |
|
|
| end |
| end |
|
|
| end OAI |
|
|
