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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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.TypeCounts
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointPairClassWindows
open MatrixMultiplication.Foundation
attribute [local instance] Classical.propDecidable
variable {I A C : Type*} [Fintype I] [Fintype A] [Fintype C]
[DecidableEq I] [DecidableEq A] [DecidableEq C]
abbrev Fiber (k : I → C) (c : C) := {i : I // k i = c}
def fiberWord (z : I → A) (k : I → C) (c : C) : Fiber k c → A :=
fun i => z i.val
omit [Fintype A] [Fintype C] [DecidableEq I] in
theorem fiber_population (z : I → A) (k : I → C) (f : A → C)
(hf : ∀ i, f (z i) = k i) (c : C) (a : A) :
wordPopulation (fiberWord z k c) a =
if f a = c then wordPopulation z a else 0 := by
classical
by_cases ha : f a = c
· rw [ite_eq_left ha]
unfold wordPopulation
apply Fintype.card_congr
exact
{ toFun := fun i => ⟨i.val.val, i.property⟩
invFun := fun i => ⟨⟨i.val, by rw [← hf i.val, i.property, ha]⟩, i.property⟩
left_inv := fun ⟨⟨i, hi⟩, hia⟩ => rfl
right_inv := fun ⟨i, hia⟩ => rfl }
· rw [ite_eq_right ha]
let : IsEmpty {i : Fiber k c // fiberWord z k c i = a} :=
⟨fun i => ha (calc
f a = f (z i.val.val) := congrArg f i.property.symm
_ = k i.val.val := hf _
_ = c := i.val.property)⟩
exact Fintype.card_eq_zero
omit [Fintype C] [DecidableEq I] in
theorem fiber_card (k : I → C) (c : C) :
Fintype.card (Fiber k c) = wordPopulation k c := rfl
def conditionalCenter (f : A → C) (c : C) (q : A → ℝ) (mass : ℝ) : A → ℝ :=
fun a => if f a = c then q a / mass else 0
omit [Fintype A] [Fintype C] [DecidableEq I] in
theorem fiber_window (z : I → A) (k : I → C) (f : A → C)
(hf : ∀ i, f (z i) = k i) (c : C) (q : A → ℝ) (mass η : ℝ)
(hmass : 0 < mass) (hη : 0 ≤ η)
(hclass : (Fintype.card (Fiber k c) : ℝ) = (Fintype.card I : ℝ) * mass)
(hwindow : ∀ a, |(wordPopulation z a : ℝ) / Fintype.card I - q a| ≤ η) :
∀ a, |(wordPopulation (fiberWord z k c) a : ℝ) / Fintype.card (Fiber k c) -
conditionalCenter f c q mass a| ≤ η / mass := by
intro a
rw [fiber_population z k f hf c a]
by_cases ha : f a = c
· simp only [ha, ite_true, conditionalCenter]
rw [hclass]
have hid : (wordPopulation z a : ℝ) / ((Fintype.card I : ℝ) * mass) - q a / mass =
((wordPopulation z a : ℝ) / Fintype.card I - q a) / mass := by
rw [sub_div, div_div]
rw [hid, abs_div, abs_of_pos hmass]
exact div_le_div_of_nonneg_right (hwindow a) hmass.le
· simp only [ha, ite_false, Nat.cast_zero, zero_div, conditionalCenter, sub_zero, abs_zero]
exact div_nonneg hη hmass.le
omit [Fintype A] [Fintype C] [DecidableEq I] in
theorem fiber_count_window (z : I → A) (k : I → C) (f : A → C)
(hf : ∀ i, f (z i) = k i) (c : C) (q : A → ℝ) (η : ℝ) (hη : 0 ≤ η)
(hwindow : ∀ a,
|(wordPopulation z a : ℝ) - (Fintype.card I : ℝ) * q a| ≤ (Fintype.card I : ℝ) * η) :
∀ a, |(wordPopulation (fiberWord z k c) a : ℝ) -
(Fintype.card I : ℝ) * (if f a = c then q a else 0)| ≤ (Fintype.card I : ℝ) * η := by
intro a
rw [fiber_population z k f hf c a]
by_cases ha : f a = c
· simpa only [ha, ite_true] using hwindow a
· simp only [ha, ite_false, Nat.cast_zero, mul_zero, sub_zero, abs_zero]
exact mul_nonneg (Nat.cast_nonneg _) hη
end MatrixMultiplication.JointPairClassWindows
end
end OAI
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