File size: 6,102 Bytes
92416a2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import Mathlib
import OAI.Computability.VertexCover.Reduction.PrivatePairProjectionRight
namespace OAI
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
namespace VertexCover.LabelCover
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators
noncomputable def otherWeightProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
product Φ.weightProjection Φ.weightProjection_commute (J.erase j).toList
theorem privateSeed_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateSeedProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateSeedProjection J j f) :=
Φ.condition_weightProjection_commute (Φ.privateSeedKey J j) k f
theorem privateSeed_otherWeight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateSeedProjection J j (Φ.otherWeightProjection J j f) =
Φ.otherWeightProjection J j (Φ.privateSeedProjection J j f) :=
(productMap_commute Φ.weightProjection (Φ.privateSeedProjection J j)
(fun k x => (Φ.privateSeed_weight_commute J j k x).symm) (J.erase j).toList f).symm
noncomputable def privateAverageProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
(Φ.privateSeedProjection J j).comp (Φ.otherWeightProjection J j)
(Φ.privateSeed_otherWeight_commute J j)
theorem privateAverage_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateAverageProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateAverageProjection J j f) := by
change Φ.privateSeedProjection J j
(productMap Φ.weightProjection (J.erase j).toList (Φ.weightProjection k f)) = _
rw [productMap_commute Φ.weightProjection (Φ.weightProjection k)
(fun l => Φ.weightProjection_commute l k), Φ.privateSeed_weight_commute]
rfl
noncomputable def privateProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
(Φ.weightProjection j).residual.comp (Φ.privateAverageProjection J j)
(residual_commute _ _ (fun f => (Φ.privateAverage_weight_commute J j j f).symm))
theorem privateProjection_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateProjection J j f) := by
change (Φ.weightProjection j).residual
(Φ.privateAverageProjection J j (Φ.weightProjection k f)) = _
rw [Φ.privateAverage_weight_commute, residual_commute _ _ (Φ.weightProjection_commute j k)]
rfl
theorem privateProjection_group_le (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy (Φ.privateProjection J j (Φ.weightGroup j f)) ≤
(Φ.restrictionForm d).energy (Φ.privateProjection J j f) :=
finDifference_retained_le Φ.weightProjection Φ.weightProjection_commute
(Φ.privateProjection J j) (fun k x => (Φ.privateProjection_weight_commute J j k x).symm) j f
theorem privateProjection_group_eq (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateProjection J j (Φ.weightGroup j f) =
Φ.privateAverageProjection J j (Φ.weightGroup j f) := by
change (Φ.weightProjection j).residual
(Φ.privateAverageProjection J j (Φ.weightGroup j f)) = _
rw [residual_commute _ _ (fun x => (Φ.privateAverage_weight_commute J j j x).symm),
Φ.weightGroup_centered]
theorem privateAverage_loss_le (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy ((Φ.privateAverageProjection J j).residual f) ≤
(∑ k ∈ J.erase j, (Φ.restrictionForm d).energy ((Φ.weightProjection k).residual f)) +
∑ e : PositionPair d, (Φ.restrictionForm d).energy ((Φ.privatePairProjection J j e).residual f) := by
have hc := comp_loss_le (Φ.privateSeedProjection J j) (Φ.otherWeightProjection J j)
(Φ.privateSeed_otherWeight_commute J j) f
have hw := product_loss_finset Φ.weightProjection Φ.weightProjection_commute (J.erase j) f
have hs := residual_le_of_nested (Φ.privateSeedProjection J j)
(product (Φ.privatePairProjection J j) (Φ.privatePairProjection_commute J j) Finset.univ.toList)
(Φ.privateSeedProjection_product J j) f
have hsp := product_loss_finset (Φ.privatePairProjection J j)
(Φ.privatePairProjection_commute J j) Finset.univ f
change (Φ.restrictionForm d).energy ((Φ.privateAverageProjection J j).residual f) ≤ _ at hc
change (Φ.restrictionForm d).energy ((Φ.otherWeightProjection J j).residual f) ≤ _ at hw
linarith
theorem privateProjection_group_lower (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy (Φ.weightGroup j f) ≤
(Φ.restrictionForm d).energy (Φ.privateProjection J j f) +
(∑ k ∈ J.erase j, (Φ.restrictionForm d).energy
((Φ.weightProjection k).residual (Φ.weightGroup j f))) +
∑ e : PositionPair d, (Φ.restrictionForm d).energy
((Φ.privatePairProjection J j e).residual (Φ.weightGroup j f)) := by
have he := (Φ.privateAverageProjection J j).pythagoras (Φ.weightGroup j f)
have hl := Φ.privateAverage_loss_le J j (Φ.weightGroup j f)
have hr := Φ.privateProjection_group_le J j f
rw [Φ.privateProjection_group_eq] at hr
linarith
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
|