| 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 |
|
|
