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f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityControls
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityScaling
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointPairMixtureControls
open JointCompatibilityControls JointCompatibilityScaling
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {C : Type*} [Fintype C] [DecidableEq C]
{Shape A B : C → Type*}
[∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)]
[∀ c, Fintype (B c)] [∀ c, DecidableEq (B c)]
def pairMixture (base : ∀ c, Shape c → ℕ)
(left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ)
(c : C) (ab : A c × B c) : ℝ :=
(∑ u, (base c u : ℝ) * left c u ab.1 * right c u ab.2) / baseSize base c
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (B c)] in
theorem sum_first_fiber (c : C) (q : A c × B c → ℝ) (a : A c) :
(∑ ab : {ab : A c × B c // ab.1 = a}, q ab.val) = ∑ b, q (a, b) := by
let e : B c ≃ {ab : A c × B c // ab.1 = a} :=
{ toFun := fun b => ⟨(a, b), rfl⟩
invFun := fun ab => ab.val.2
left_inv := fun _ => rfl
right_inv := fun ab => by apply Subtype.ext; exact Prod.ext ab.property.symm rfl }
exact (e.sum_comp (fun ab => q ab.val)).symm
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (A c)] in
theorem sum_second_fiber (c : C) (q : A c × B c → ℝ) (b : B c) :
(∑ ab : {ab : A c × B c // ab.2 = b}, q ab.val) = ∑ a, q (a, b) := by
let e : A c ≃ {ab : A c × B c // ab.2 = b} :=
{ toFun := fun a => ⟨(a, b), rfl⟩
invFun := fun ab => ab.val.1
left_inv := fun _ => rfl
right_inv := fun ab => by apply Subtype.ext; exact Prod.ext rfl ab.property.symm }
exact (e.sum_comp (fun ab => q ab.val)).symm
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] in
theorem base_eq_zero_of_size_eq_zero (base : ∀ c, Shape c → ℕ)
(c : C) (hz : baseSize base c = 0) (u : Shape c) : base c u = 0 :=
Nat.eq_zero_of_le_zero ((Finset.single_le_sum (fun _ _ => Nat.zero_le _)
(Finset.mem_univ u)).trans_eq hz)
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] in
theorem totalCenterCount_eq_zero_of_size_eq_zero (base : ∀ c, Shape c → ℕ)
(law : ∀ c, Shape c → A c → ℝ) (c : C) (hz : baseSize base c = 0) (a : A c) :
totalCenterCount base law c a = 0 := by
simp only [totalCenterCount, base_eq_zero_of_size_eq_zero base c hz,
Nat.cast_zero, zero_mul, Finset.sum_const_zero]
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)]
[∀ c, DecidableEq (B c)] in
theorem pairMixture_left_balance (base : ∀ c, Shape c → ℕ)
(left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ)
(hnormalized : ∀ c u, 0 < base c u → ∑ b, right c u b = 1)
(c : C) (a : A c) :
(baseSize base c : ℝ) *
(∑ ab : {ab : A c × B c // ab.1 = a}, pairMixture base left right c ab.val) =
totalCenterCount base left c a := by
rw [sum_first_fiber]
simp only [pairMixture, ← Finset.sum_div]
have hnum : (∑ b : B c, ∑ u : Shape c,
(base c u : ℝ) * left c u a * right c u b) = totalCenterCount base left c a := by
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro u _
by_cases hu : base c u = 0
· simp [hu]
· rw [← Finset.mul_sum, hnormalized c u (Nat.pos_of_ne_zero hu), mul_one]
rw [hnum]
by_cases hz : baseSize base c = 0
· rw [totalCenterCount_eq_zero_of_size_eq_zero base left c hz a]
simp
· exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hz)
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)]
[∀ c, DecidableEq (A c)] in
theorem pairMixture_right_balance (base : ∀ c, Shape c → ℕ)
(left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ)
(hnormalized : ∀ c u, 0 < base c u → ∑ a, left c u a = 1)
(c : C) (b : B c) :
(baseSize base c : ℝ) *
(∑ ab : {ab : A c × B c // ab.2 = b}, pairMixture base left right c ab.val) =
totalCenterCount base right c b := by
rw [sum_second_fiber]
simp only [pairMixture, ← Finset.sum_div]
have hnum : (∑ a : A c, ∑ u : Shape c,
(base c u : ℝ) * left c u a * right c u b) = totalCenterCount base right c b := by
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro u _
by_cases hu : base c u = 0
· simp [hu]
· rw [← Finset.sum_mul, ← Finset.mul_sum,
hnormalized c u (Nat.pos_of_ne_zero hu), mul_one]
rw [hnum]
by_cases hz : baseSize base c = 0
· rw [totalCenterCount_eq_zero_of_size_eq_zero base right c hz b]
simp
· exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hz)
end MatrixMultiplication.JointPairMixtureControls
end
end OAI
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