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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | 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; ring]
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; simp [hdiag])]
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; simp [hdiag])]
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
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