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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 | import OAI.Combinatorics.MatchingPSD.TablePolynomials
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9
namespace MatchingSmoothing
open scoped BigOperators
attribute [local instance] Classical.propDecidable
section Averages
variable {X : Type u_1} {Y : Type u_2} [Fintype X] [Fintype Y]
lemma predicateAverage_equiv (e : X ≃ Y) (P : X → Prop) (Q : Y → Prop)
(he : ∀ x, P x ↔ Q (e x)) (f : Y → ℝ) :
predicateAverage P (f ∘ e) = predicateAverage Q f := by
classical
have hc : Fintype.card {x // P x} = Fintype.card {y // Q y} :=
Fintype.card_congr (e.subtypeEquiv he)
rw [predicateAverage_apply, predicateAverage_apply, hc]
congr 1
exact Fintype.sum_equiv e _ _ (fun x => by simp only [Function.comp_apply, he])
lemma predicateAverage_eq_expect (P : X → Prop) (f : X → ℝ) :
predicateAverage P f = 𝔼 x : {x // P x}, f x.1 := by
rw [predicateAverage_apply, Fintype.expect_eq_sum_div_card]
congr 1
rw [← Finset.sum_filter]
exact Finset.sum_subtype (Finset.univ.filter P) (by simp) f
end Averages
noncomputable def lowProjection (h d : ℕ) : Module.End ℝ (Pairing (Vertices h) → ℝ) :=
(matchingEquiv h).toLinearMap ∘ₗ (polynomialSpace h d).starProjection.toLinearMap ∘ₗ
(matchingEquiv h).symm.toLinearMap
lemma lowProjection_mem (h d : ℕ) (f : Pairing (Vertices h) → ℝ) :
lowProjection h d f ∈ lowDegree Pairing.edgeCoord d := by
exact ((polynomialSpace h d).orthogonalProjectionOnto ((matchingEquiv h).symm f)).property
lemma lowProjection_equivariant (h d : ℕ) (p : Equiv.Perm (Vertices h))
(f : Pairing (Vertices h) → ℝ) :
lowProjection h d (Pairing.act p f) = Pairing.act p (lowProjection h d f) := by
have he := starProjection_equivariant (polynomialSpace h d) (relabelIso p)
(polynomialSpace_invariant p) (by simpa only [relabelIso_symm] using
(polynomialSpace_invariant (d := d) p⁻¹)) ((matchingEquiv h).symm f)
exact congrArg (matchingEquiv h) he.symm
lemma predicateAverage_relabel {V : Type u_3} [Fintype V] [DecidableEq V]
(z : V → Fin 3) (t : Fin 6 → ℕ) (p : Equiv.Perm V) (f : Pairing V → ℝ) :
predicateAverage (fun M => M.table z = t) (fun M => f (Pairing.relabel p M)) =
predicateAverage (fun M => M.table (z ∘ p.symm) = t) f := by
apply predicateAverage_equiv (Pairing.relabelEquiv p)
intro M
change M.table z = t ↔ (Pairing.relabel p M).table (z ∘ p.symm) = t
rw [Pairing.table_relabel]
have hz : (z ∘ p.symm) ∘ p = z := by funext a; simp
rw [hz]
lemma expect_perm_colorings {V : Type u_4} {C : Type u_5} [Fintype V] [DecidableEq V]
[Fintype C] [DecidableEq C] (z : V → C) (φ : (V → C) → ℝ) :
(𝔼 p : Equiv.Perm V, φ (z ∘ p)) =
𝔼 w : Colorings V C (histogram z), φ w.1 := by
have : Nonempty (Colorings V C (histogram z)) := ⟨⟨z,rfl⟩⟩
rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card]
apply (div_eq_div_iff (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
(Nat.cast_ne_zero.mpr Fintype.card_ne_zero)).mpr
simpa only [mul_comm] using mean_perm_comp z φ
noncomputable def tableEnergy {V : Type u_6} [Fintype V] (s : Fin 3 → ℕ)
(t : Fin 6 → ℕ) (f : Pairing V → ℝ) : ℝ :=
𝔼 z : Colorings V (Fin 3) s, (predicateAverage (fun M => M.table z.1 = t) f)^2
lemma fiberAverage_eq_predicateAverage {X : Type u_7} {Z : Type u_8} [Fintype X] [Fintype Z]
(P : X → Z → Prop) (f : X → ℝ) (z : Z) :
fiberAverage P f z = predicateAverage (fun x => P x z) f := by
simp only [fiberAverage, conditionalAverage, incidence, predicateAverage_apply,
ite_mul, one_mul, zero_mul]
lemma fiberEnergy_eq {h : ℕ} (f₀ : Vertices h → Fin 3)
(f : Pairing (Vertices h) → ℝ) :
(∑ z, fiberAverage (tableRelation f₀) f z ^ 2) /
Fintype.card (Colorings (Vertices h) (Fin 3) (histogram f₀)) =
tableEnergy (histogram f₀) (threeTable f₀) f := by
simp_rw [fiberAverage_eq_predicateAverage]
rw [tableEnergy, Fintype.expect_eq_sum_div_card]
rfl
lemma permutation_energy_eq {V : Type u_9} [Fintype V] [DecidableEq V]
(z : V → Fin 3) (t : Fin 6 → ℕ) (f : Pairing V → ℝ) :
(𝔼 p : Equiv.Perm V, (predicateAverage (fun M => M.table z = t)
(fun M => f (Pairing.relabel p M)))^2) = tableEnergy (histogram z) t f := by
simp_rw [predicateAverage_relabel]
calc
_ = 𝔼 p : Equiv.Perm V,
(predicateAverage (fun M => M.table (z ∘ p) = t) f)^2 := by
apply Fintype.expect_equiv (Equiv.inv (Equiv.Perm V))
intro p
rfl
_ = _ := expect_perm_colorings z (fun w =>
(predicateAverage (fun M => M.table w = t) f)^2)
lemma uniform_permutation_smoothing (K : ℕ) :
∃ d H : ℕ, 0 < d ∧ ∀ h : ℕ, H ≤ h →
∀ (z : Vertices h → Fin 3), (∀ i, 0 < histogram z i) →
∀ (M₀ : Pairing (Vertices h)) (f : Pairing (Vertices h) → ℝ),
(𝔼 p : Equiv.Perm (Vertices h),
(predicateAverage (fun M => M.table z = M₀.table z)
(fun M => (f - lowProjection h d f) (Pairing.relabel p M)))^2) ≤
((2*h : ℝ)^(2*K))⁻¹ * (𝔼 M, (f M)^2) := by
obtain ⟨d,H,hd,hbound⟩ := uniform_table_smoothing K
refine ⟨d,H,hd,?_⟩
intro h hh z hz M₀ f
obtain ⟨p₀, hp₀⟩ := Pairing.exists_relabel (standardPairing (H := Fin h)) M₀
let f₀ := z ∘ p₀
have hhist : histogram f₀ = histogram z := histogram_comp_perm z p₀
have ht : threeTable f₀ = M₀.table z := by
rw [← hp₀, Pairing.table_relabel, standardPairing_table]
have hb := hbound h hh f₀ (by simpa only [hhist] using hz) ((matchingEquiv h).symm f)
have heg : matchingEquiv h ((matchingEquiv h).symm f -
polynomialProjection h d ((matchingEquiv h).symm f)) = f - lowProjection h d f := by
simp only [map_sub, LinearEquiv.apply_symm_apply]
rfl
rw [fiberEnergy_eq, heg, ht, hhist] at hb
rw [permutation_energy_eq, Fintype.expect_eq_sum_div_card]
exact hb
end MatchingSmoothing
end OAI
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