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e5a97c6 | 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 | import OAI.Combinatorics.MatchingPSD.TableMoments
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6
namespace MatchingSmoothing
open Matrix
section FiniteSums
variable {G : Type u_1} {X : Type u_2} [Fintype G] [Group G] [Fintype X] [MulAction G X]
omit [Fintype G] in
lemma sum_smul_eq (g : G) (f : X → ℝ) : ∑ x, f (g • x) = ∑ x, f x :=
Equiv.sum_comp (MulAction.toPerm g) f
omit [Fintype X] in
lemma sum_group_smul_eq (x₀ : X) (ht : ∀ x, ∃ g : G, g • x₀ = x)
(f : X → ℝ) (x : X) : ∑ g : G, f (g • x) = ∑ g : G, f (g • x₀) := by
obtain ⟨h, rfl⟩ := ht x
simp_rw [← mul_smul]
exact Equiv.sum_comp (Equiv.mulRight h) (fun g => f (g • x₀))
lemma group_smul_sum_card (x₀ : X) (ht : ∀ x, ∃ g : G, g • x₀ = x)
(f : X → ℝ) :
(Fintype.card G : ℝ) * ∑ x, f x =
(Fintype.card X : ℝ) * ∑ g : G, f (g • x₀) := by
calc
_ = ∑ g : G, ∑ x, f (g • x) := by simp only [sum_smul_eq, Finset.sum_const,
Finset.card_univ, nsmul_eq_mul]
_ = ∑ x, ∑ g : G, f (g • x) := Finset.sum_comm
_ = _ := by simp_rw [sum_group_smul_eq x₀ ht, Finset.sum_const,
Finset.card_univ, nsmul_eq_mul]
omit [Fintype G] in
lemma invariant_kernel_sum (x₀ : X) (ht : ∀ x, ∃ g : G, g • x₀ = x)
(K : X → X → ℝ) (hK : ∀ (g : G) x y, K (g • x) (g • y) = K x y) :
(∑ x, ∑ y, K x y) = (Fintype.card X : ℝ) * ∑ y, K x₀ y := by
have he (x : X) : ∑ y, K x y = ∑ y, K x₀ y := by
obtain ⟨g, rfl⟩ := ht x
rw [← sum_smul_eq g (fun y => K (g • x₀) y)]
simp_rw [hK]
simp_rw [he, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
lemma invariant_kernel_double_sum_group (x₀ : X)
(ht : ∀ x, ∃ g : G, g • x₀ = x)
(K : X → X → ℝ) (hK : ∀ (g : G) x y, K (g • x) (g • y) = K x y) :
(Fintype.card G : ℝ) * (∑ x, ∑ y, K x y) =
(Fintype.card X : ℝ) ^ 2 * ∑ g : G, K x₀ (g • x₀) := by
rw [invariant_kernel_sum x₀ ht K hK]
have h := group_smul_sum_card x₀ ht (K x₀)
nlinarith [h]
end FiniteSums
section KernelTrace
variable {G : Type u_3} {X : Type u_4} [Fintype G] [Group G] [Fintype X] [DecidableEq X]
[MulAction G X]
omit [DecidableEq X] in
lemma trace_normalized_kernel_sq (K : Matrix X X ℝ) (hs : ∀ x y, K x y = K y x) :
Matrix.trace ((Matrix.of fun x y => K x y / Fintype.card X) *
(Matrix.of fun x y => K x y / Fintype.card X)) =
(∑ x, ∑ y, K x y ^ 2) / (Fintype.card X : ℝ) ^ 2 := by
simp only [Matrix.trace, Matrix.diag, Matrix.mul_apply, Matrix.of_apply, Finset.sum_div]
apply Finset.sum_congr rfl
intro x _
apply Finset.sum_congr rfl
intro y _
rw [hs y x]
ring
omit [DecidableEq X] in
lemma trace_normalized_invariant_kernel_sq (x₀ : X)
(ht : ∀ x, ∃ g : G, g • x₀ = x)
(K : Matrix X X ℝ) (hs : ∀ x y, K x y = K y x)
(hi : ∀ (g : G) x y, K (g • x) (g • y) = K x y) :
Matrix.trace ((Matrix.of fun x y => K x y / Fintype.card X) *
(Matrix.of fun x y => K x y / Fintype.card X)) =
(∑ g : G, K x₀ (g • x₀) ^ 2) / (Fintype.card G : ℝ) := by
have : Nonempty X := ⟨x₀⟩
rw [trace_normalized_kernel_sq K hs]
have h := invariant_kernel_double_sum_group x₀ ht
(fun x y => K x y ^ 2) (fun g x y => congrArg (fun a : ℝ => a ^ 2) (hi g x y))
have hx : (Fintype.card X : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
have hg : (Fintype.card G : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
apply (div_eq_div_iff (pow_ne_zero _ hx) hg).mpr
nlinarith [h]
end KernelTrace
section Incidence
variable {X : Type u_5} {Z : Type u_6} [Fintype X] [Fintype Z] [DecidableEq X] [DecidableEq Z]
noncomputable def incidence (P : X → Z → Prop) : Matrix Z X ℝ :=
by classical exact fun z x => if P x z then 1 else 0
noncomputable def commonIncidence (P : X → Z → Prop) (x y : X) : ℝ :=
by classical exact (Fintype.card {z // P x z ∧ P y z} : ℝ)
omit [Fintype X] [DecidableEq X] [DecidableEq Z] in
lemma commonIncidence_eq (P : X → Z → Prop) (x y : X) :
commonIncidence P x y = ∑ z, incidence P z x * incidence P z y := by
classical
simp only [commonIncidence, Fintype.card_subtype, Finset.card_filter, Nat.cast_sum]
apply Finset.sum_congr rfl
intro z _
by_cases hx : P x z <;> by_cases hy : P y z <;> simp [incidence, hx, hy]
omit [Fintype X] [DecidableEq X] [DecidableEq Z] in
lemma commonIncidence_symm (P : X → Z → Prop) (x y : X) :
commonIncidence P x y = commonIncidence P y x := by
simp only [commonIncidence_eq, mul_comm]
noncomputable def incidenceKernel (P : X → Z → Prop) (D : ℝ) : Matrix X X ℝ :=
fun x y => (Fintype.card Z : ℝ) / D ^ 2 * commonIncidence P x y
noncomputable def incidenceOperator (P : X → Z → Prop) (D : ℝ) : Matrix X X ℝ :=
fun x y => incidenceKernel P D x y / Fintype.card X
omit [Fintype X] [DecidableEq X] [DecidableEq Z] in
lemma incidenceKernel_symm (P : X → Z → Prop) (D : ℝ) (x y : X) :
incidenceKernel P D x y = incidenceKernel P D y x := by
unfold incidenceKernel
rw [commonIncidence_symm P x y]
omit [DecidableEq X] [DecidableEq Z] in
lemma incidenceOperator_eq (P : X → Z → Prop) (D : ℝ) :
incidenceOperator P D =
((Fintype.card Z : ℝ) / (D ^ 2 * Fintype.card X)) •
((incidence P).transpose * incidence P) := by
ext x y
simp only [incidenceOperator, incidenceKernel, commonIncidence_eq, Matrix.smul_apply,
smul_eq_mul, Matrix.mul_apply, Matrix.transpose_apply]
ring
omit [DecidableEq X] [DecidableEq Z] in
lemma incidenceOperator_posSemidef (P : X → Z → Prop) (D : ℝ) :
(incidenceOperator P D).PosSemidef := by
rw [incidenceOperator_eq]
apply Matrix.PosSemidef.smul
· simpa only [Matrix.conjTranspose_eq_transpose_of_trivial] using
Matrix.posSemidef_conjTranspose_mul_self (incidence P)
· positivity
noncomputable def conditionalAverage (P : X → Z → Prop) (E : ℝ) (f : X → ℝ) (z : Z) : ℝ :=
(∑ x, incidence P z x * f x) / E
omit [DecidableEq X] [DecidableEq Z] in
lemma conditionalAverage_energy [Nonempty X] [Nonempty Z]
(P : X → Z → Prop) (D : ℝ) (hD : D ≠ 0) (f : X → ℝ) :
(∑ z, conditionalAverage P ((Fintype.card X : ℝ) * D / Fintype.card Z) f z ^ 2) /
Fintype.card Z =
(∑ x, f x * (incidenceOperator P D *ᵥ f) x) / Fintype.card X := by
have hx : (Fintype.card X : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
have hz : (Fintype.card Z : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
change _ = (f ⬝ᵥ (incidenceOperator P D *ᵥ f)) / Fintype.card X
rw [incidenceOperator_eq, Matrix.smul_mulVec, dotProduct_smul,
← Matrix.mulVec_mulVec, Matrix.dotProduct_mulVec, Matrix.vecMul_transpose]
simp only [conditionalAverage, div_pow, ← Finset.sum_div, smul_eq_mul]
change (∑ z, ((incidence P *ᵥ f) z) ^ 2) /
(((Fintype.card X : ℝ) * D) ^ 2 / (Fintype.card Z : ℝ) ^ 2) / Fintype.card Z =
((Fintype.card Z : ℝ) / (D ^ 2 * Fintype.card X)) *
(∑ z, (incidence P *ᵥ f) z * (incidence P *ᵥ f) z) / Fintype.card X
simp only [← pow_two]
field_simp
end Incidence
end MatchingSmoothing
end OAI
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