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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 | import OAI.Combinatorics.MatchingPSD.MatrixBounds
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6
namespace PerfectMatchingPSD
open Matrix MatchingSmoothing
open scoped BigOperators MatrixOrder
attribute [local instance] Classical.propDecidable
variable {R : Type u_1} {X : Type u_2} [Fintype R] [DecidableEq R] [Fintype X]
omit [Fintype R] [DecidableEq R] in
lemma matrix_expect_eq (f : X → Matrix R R ℝ) :
(𝔼 x, f x) = ((Fintype.card X : ℝ)⁻¹) • (∑ x, f x) := by
ext i j
simp [Finset.expect, NNRat.smul_def]
omit [Fintype R] [DecidableEq R] in
lemma matrix_expect_apply (f : X → Matrix R R ℝ) (u v : R) :
(𝔼 x, f x) u v = 𝔼 x, f x u v := by
rw [matrix_expect_eq, Fintype.expect_eq_sum_div_card]
simp only [Matrix.smul_apply, Matrix.sum_apply, smul_eq_mul, div_eq_inv_mul]
omit [Fintype R] [DecidableEq R] in
lemma matrix_expect_psd (f : X → Matrix R R ℝ) (hf : ∀ x, (f x).PosSemidef) :
(𝔼 x, f x).PosSemidef := by
rw [matrix_expect_eq]
exact (matrix_sum_psd Finset.univ f (fun x _ => hf x)).smul
(inv_nonneg.mpr (Nat.cast_nonneg _))
lemma trace_mul_expect (Q : Matrix R R ℝ) (f : X → Matrix R R ℝ) :
(Q * (𝔼 x, f x)).trace = 𝔼 x, (Q*f x).trace := by
rw [matrix_expect_eq, mul_smul_comm, Matrix.trace_smul,
Finset.mul_sum, Matrix.trace_sum, Fintype.expect_eq_sum_div_card]
change (Fintype.card X : ℝ)⁻¹ * _ = _ / Fintype.card X
rw [div_eq_inv_mul]
lemma expect_abs_bound (f : X → ℝ) {b : ℝ} (hb : 0 ≤ b) (hf : ∀ x, |f x| ≤ b) :
|𝔼 x, f x| ≤ b := by
by_cases hX : Nonempty X
· let := hX
have h0 (x : X) : -b ≤ f x := (abs_le.mp (hf x)).1
have h1 (x : X) : f x ≤ b := (abs_le.mp (hf x)).2
apply abs_le.mpr
constructor
· simpa using Finset.expect_le_expect (s := Finset.univ) (fun x _ => h0 x)
· simpa using Finset.expect_le_expect (s := Finset.univ) (fun x _ => h1 x)
· have : IsEmpty X := not_nonempty_iff.mp hX
simpa using hb
section CellMatrices
variable {A : Type u_3} {B : Type u_4} {C : Type u_5} {I : Type u_6} [Fintype A] [Fintype B] [Fintype C] [Fintype I] [DecidableEq I]
noncomputable def extensionMatrix (i : I)
(Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(M : Pairing ((A ⊕ B) ⊕ C)) : Matrix R R ℝ :=
𝔼 N : {j : I // j ≠ i} → Pairing C, Bf (extendLocal i M N)
noncomputable def cellMatrix (i : I)
(Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(x : Fin 3 → ℝ) : Matrix R R ℝ :=
𝔼 M : {M : Pairing ((A ⊕ B) ⊕ C) // crossVector threeColor M = x}, extensionMatrix i Bf M.1
omit [Fintype R] [DecidableEq R] in
lemma cellMatrix_apply (i : I) (Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(x : Fin 3 → ℝ) (u v : R) :
cellMatrix i Bf x u v = cellAverage i (fun M => Bf M u v) x := by
rw [cellAverage, predicateAverage_eq_expect]
simp only [cellMatrix, extensionMatrix, extensionAverage, matrix_expect_apply]
apply Finset.expect_congr (by ext; simp)
intro M _
rfl
omit [Fintype R] [DecidableEq R] in
lemma cellMatrix_psd (i : I) (Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(hB : ∀ M, (Bf M).PosSemidef) (x : Fin 3 → ℝ) :
(cellMatrix i Bf x).PosSemidef := by
apply matrix_expect_psd
intro M
apply matrix_expect_psd
intro N
exact hB _
omit [Fintype R] [DecidableEq R] in
lemma cellMatrix_star (i j : I) (Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(x : ℝ) : cellMatrix i Bf ![x,0,0] = cellMatrix j Bf ![x,0,0] := by
ext u v
simp only [cellMatrix_apply, cellAverage_star]
lemma trace_cellMatrix (i : I) (Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(Q : Matrix R R ℝ) (x : Fin 3 → ℝ) :
(Q*cellMatrix i Bf x).trace = cellAverage i (fun M => (Q*Bf M).trace) x := by
rw [cellAverage, predicateAverage_eq_expect, cellMatrix, trace_mul_expect]
apply Finset.expect_congr (by ext; simp)
intro M _
exact trace_mul_expect Q _
omit [Fintype A] [Fintype B] in
lemma extensionAverage_sq_bound (i : I)
(Bf : Pairing ((A ⊕ B) ⊕ (I × C)) → Matrix R R ℝ)
(hB : ∀ M, (Bf M).PosSemidef) {b : ℝ} (hb : 0 ≤ b)
(htr : ∀ M, (Bf M).trace ≤ b) (u v : R) (M : Pairing ((A ⊕ B) ⊕ C)) :
(extensionAverage i (fun N => Bf N u v) M)^2 ≤ b^2 := by
have he := expect_abs_bound (fun N => Bf (extendLocal i M N) u v) hb
(fun N => (posSemidef_entry_abs_le_trace (hB _) u v).trans (htr _))
have hex : extensionAverage i (fun N => Bf N u v) M =
𝔼 N : {j : I // j ≠ i} → Pairing C, Bf (extendLocal i M N) u v := by
apply Finset.expect_congr (by ext; simp)
intro N _
rfl
rw [hex]
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hb).mpr he
end CellMatrices
end PerfectMatchingPSD
end OAI
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