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2f887db | 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 | import Mathlib.Algebra.BigOperators.Expect
import Mathlib.LinearAlgebra.Prod
import OAI.Computability.PerfectCompleteness.Algebra.BilinearGramCompressionLemmas
import OAI.Computability.PerfectCompleteness.Algebra.UniformLinearImage
import OAI.Computability.UniqueGames.Games.FinishBoundsLemmas
namespace OAI
section
namespace PerfectCompleteness.MatrixRowQuotientLaw
noncomputable section
open scoped BigOperators
open UniqueGamesTheorem.Foundations.Games
open MatrixRowQuotient
variable {𝕜 H K : Type*} [Field 𝕜]
[AddCommGroup H] [Module 𝕜 H] [AddCommGroup K] [Module 𝕜 K]
def projectMatrixLinear (W : Submodule 𝕜 H) :
(Module.Dual 𝕜 H →ₗ[𝕜] K) →ₗ[𝕜] (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) where
toFun := projectMatrix W
map_add' X Y := by
apply LinearMap.ext
intro q
rfl
map_smul' a X := by
apply LinearMap.ext
intro q
rfl
@[simp] theorem projectMatrixLinear_apply (W : Submodule 𝕜 H)
(X : Module.Dual 𝕜 H →ₗ[𝕜] K) :
projectMatrixLinear W X = projectMatrix W X := rfl
def jointProjection (W : Submodule 𝕜 H) :
((Module.Dual 𝕜 H →ₗ[𝕜] K) × H) →ₗ[𝕜]
((Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) × (H ⧸ W)) :=
(projectMatrixLinear W).prodMap W.mkQ
@[simp] theorem jointProjection_apply (W : Submodule 𝕜 H)
(p : (Module.Dual 𝕜 H →ₗ[𝕜] K) × H) :
jointProjection W p = (projectMatrix W p.1, W.mkQ p.2) := rfl
theorem jointProjection_surjective (W : Submodule 𝕜 H) :
Function.Surjective (jointProjection (K := K) W) := by
rintro ⟨X, h⟩
obtain ⟨Y, hY⟩ := projectMatrix_surjective W X
obtain ⟨g, hg⟩ := W.mkQ_surjective h
exact ⟨(Y, g), Prod.ext hY hg⟩
theorem uniform_product {A B : Type*} [Fintype A] [Fintype B]
[Nonempty A] [Nonempty B] :
(FiniteDistribution.uniform A).product (FiniteDistribution.uniform B) =
FiniteDistribution.uniform (A × B) := by
apply FiniteDistribution.eq_of_weight_eq
intro p
change (1 / (Fintype.card A : ℝ)) * (1 / (Fintype.card B : ℝ)) =
1 / (Fintype.card (A × B) : ℝ)
simp only [Fintype.card_prod, Nat.cast_mul, one_div, mul_inv]
variable (W : Submodule 𝕜 H)
[Fintype H] [Fintype (H ⧸ W)]
[Fintype (Module.Dual 𝕜 H →ₗ[𝕜] K)]
[Fintype (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K)]
theorem uniform_jointProjection :
(FiniteDistribution.uniform ((Module.Dual 𝕜 H →ₗ[𝕜] K) × H)).pushforward
(jointProjection W) =
FiniteDistribution.uniform ((Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) × (H ⧸ W)) :=
UniformLinearImage.uniform_pushforward_linearMap
(jointProjection W) (jointProjection_surjective W)
theorem independent_uniform_jointProjection :
((FiniteDistribution.uniform (Module.Dual 𝕜 H →ₗ[𝕜] K)).product
(FiniteDistribution.uniform H)).pushforward (jointProjection W) =
(FiniteDistribution.uniform (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K)).product
(FiniteDistribution.uniform (H ⧸ W)) := by
rw [uniform_product, uniform_product]
exact uniform_jointProjection W
private theorem expect_prod {A B : Type*} [Fintype A] [Fintype B]
(f : A × B → ℝ) :
(𝔼 p : A × B, f p) = 𝔼 a : A, 𝔼 b : B, f (a, b) := by
simpa only [Finset.univ_product_univ] using
Finset.expect_product (Finset.univ : Finset A) (Finset.univ : Finset B) f
theorem expect_project_matrix_increment
(f : (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) → (H ⧸ W) → ℝ) :
(𝔼 X : Module.Dual 𝕜 H →ₗ[𝕜] K, 𝔼 h : H,
f (projectMatrix W X) (W.mkQ h)) =
𝔼 X : Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K, 𝔼 h : H ⧸ W, f X h := by
have he := congrArg
(fun μ : FiniteDistribution ((Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) × (H ⧸ W)) =>
μ.expectation (fun p => f p.1 p.2)) (uniform_jointProjection W)
rw [FiniteDistribution.expectation_pushforward,
FiniteDistribution.expectation_uniform, FiniteDistribution.expectation_uniform] at he
have he' :
(𝔼 p : (Module.Dual 𝕜 H →ₗ[𝕜] K) × H,
f (projectMatrix W p.1) (W.mkQ p.2)) =
𝔼 p : (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) × (H ⧸ W), f p.1 p.2 := by
simpa only [Fintype.expect_eq_sum_div_card, jointProjection_apply] using he
simpa only [expect_prod] using he'
theorem rankOne_pair_law (a : K) :
((FiniteDistribution.uniform (Module.Dual 𝕜 H →ₗ[𝕜] K)).product
(FiniteDistribution.uniform H)).pushforward
(fun p => (projectMatrix W p.1,
projectMatrix W (p.1 + (Module.Dual.eval 𝕜 H p.2).smulRight a))) =
((FiniteDistribution.uniform (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K)).product
(FiniteDistribution.uniform (H ⧸ W))).pushforward
(fun p => (p.1, p.1 + (Module.Dual.eval 𝕜 (H ⧸ W) p.2).smulRight a)) := by
have he := congrArg
(fun μ : FiniteDistribution ((Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) × (H ⧸ W)) =>
μ.pushforward (fun p =>
(p.1, p.1 + (Module.Dual.eval 𝕜 (H ⧸ W) p.2).smulRight a)))
(independent_uniform_jointProjection W)
rw [FiniteDistribution.pushforward_comp] at he
simpa only [jointProjection_apply, project_rankOne] using he
theorem expect_project_rankOne_pair (a : K)
(f : (Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) →
(Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K) → ℝ) :
(𝔼 X : Module.Dual 𝕜 H →ₗ[𝕜] K, 𝔼 h : H,
f (projectMatrix W X)
(projectMatrix W (X + (Module.Dual.eval 𝕜 H h).smulRight a))) =
𝔼 X : Module.Dual 𝕜 (H ⧸ W) →ₗ[𝕜] K, 𝔼 h : H ⧸ W,
f X (X + (Module.Dual.eval 𝕜 (H ⧸ W) h).smulRight a) := by
simpa only [project_rankOne] using expect_project_matrix_increment W
(fun X h => f X (X + (Module.Dual.eval 𝕜 (H ⧸ W) h).smulRight a))
end
end PerfectCompleteness.MatrixRowQuotientLaw
end
end OAI
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