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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityRateLimit
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityControls
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.GroupedEntropyPartition
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointProjectedGroupEntropy
open MatrixMultiplication.Foundation JointCompatibilityIncidence
open JointCompatibilityScaling JointCompatibilityControls JointCompatibilityRateLimit
open JointGroupedEntropyPartition
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
section FiniteGroups
variable {U K A : Type*} [Fintype U] [Fintype K] [Fintype A]
theorem groupMass_designated_some (p : U → ℝ) (d : U → Prop) (u : U) :
groupMass p (designatedGroup d) (some u) = if d u then p u else 0 := by
simp only [groupMass, designatedGroup_eq_some]
by_cases hd : d u <;> simp [hd]
theorem weighted_groupLaw_designated_some (p : U → ℝ) (d : U → Prop)
(law : U → A → ℝ) (u : U) :
groupMass p (designatedGroup d) (some u) *
finiteEntropy (groupLaw p (designatedGroup d) law (some u)) =
(if d u then p u else 0) * finiteEntropy (law u) := by
by_cases hd : d u
· rw [groupMass_designated_some, ite_eq_left hd]
by_cases hp : p u = 0
· simp [hp]
· have hl : groupLaw p (designatedGroup d) law (some u) = law u := by
funext a
simp [groupLaw, groupMass_designated_some, designatedGroup_eq_some, hd, hp]
rw [hl]
· rw [groupMass_designated_some, ite_eq_right hd]
simp
theorem groupedEntropy_counts_eq_total_mul (n : U → ℕ) (group : U → K)
(law : U → A → ℝ) :
groupedEntropy (fun u => (n u : ℝ)) group law =
(∑ u, n u : ℕ) *
groupedEntropy (fun u => (n u : ℝ) / (∑ v, n v : ℕ)) group law := by
by_cases hz : (∑ u, n u) = 0
· have hn (u : U) : n u = 0 := by
have hle : n u ≤ ∑ v, n v :=
Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ u)
rw [hz] at hle
exact Nat.eq_zero_of_le_zero hle
simp [hn, groupedEntropy, groupMass]
· have hN : ((∑ u, n u : ℕ) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hz
have hf : (fun u => (n u : ℝ)) =
fun u => ((∑ v, n v : ℕ) : ℝ) * ((n u : ℝ) / (∑ v, n v : ℕ)) := by
funext u
exact (mul_div_cancel₀ _ hN).symm
calc
_ = groupedEntropy
(fun u => ((∑ v, n v : ℕ) : ℝ) * ((n u : ℝ) / (∑ v, n v : ℕ)))
group law := congrArg (fun p => groupedEntropy p group law) hf
_ = _ := groupedEntropy_mul _ _ _ _
end FiniteGroups
variable {C : Type*} [Fintype C] [DecidableEq C]
{Shape A : C → Type*}
[∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)]
omit [Fintype C] [DecidableEq C] in
theorem baseGroupSize_eq_groupMass (base : ∀ c, Shape c → ℕ)
(d : ∀ c, Shape c → Prop) (g : Σ c, Option (Shape c)) :
(baseGroupSize base d g : ℝ) =
groupMass (fun u => (base g.1 u : ℝ)) (designatedGroup (d g.1)) g.2 := by
calc
(baseGroupSize base d g : ℝ) =
∑ u ∈ Finset.univ.filter (fun u => designatedGroup (d g.1) u = g.2),
(base g.1 u : ℝ) := by
rw [baseGroupSize, Nat.cast_sum]
exact (Finset.sum_subtype
(Finset.univ.filter (fun u => designatedGroup (d g.1) u = g.2))
(by simp) (fun u => (base g.1 u : ℝ))).symm
_ = ∑ u, if designatedGroup (d g.1) u = g.2 then (base g.1 u : ℝ) else 0 := by
rw [Finset.sum_filter]
_ = _ := by
unfold groupMass
apply Finset.sum_congr rfl
intro u _
by_cases hu : designatedGroup (d g.1) u = g.2 <;> simp only [hu, ite_true, ite_false]
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] in
theorem residualCenter_eq_groupLaw (base : ∀ c, Shape c → ℕ)
(d : ∀ c, Shape c → Prop) (law : ∀ c, Shape c → A c → ℝ) (c : C) :
residualCenter base d law c =
groupLaw (fun u => (base c u : ℝ)) (designatedGroup (d c)) (law c) none := by
funext a
simp only [residualCenter, residualBase, Nat.cast_sum, groupLaw, groupMass,
designatedGroup_eq_none]
congr 1
· rw [← Finset.sum_filter]
exact (Finset.sum_subtype (Finset.univ.filter (fun u => ¬ d c u))
(by simp) (fun u => (base c u : ℝ) * law c u a)).symm
· rw [← Finset.sum_filter]
exact (Finset.sum_subtype (Finset.univ.filter (fun u => ¬ d c u))
(by simp) (fun u => (base c u : ℝ))).symm
omit [DecidableEq C] [∀ c, DecidableEq (A c)] in
theorem projectedReferenceEntropy_eq_sum_counts_groupedEntropy
(base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop)
(law : ∀ c, Shape c → A c → ℝ) :
projectedReferenceEntropy base d (groupCap base d law 0) =
∑ c, groupedEntropy (fun u => (base c u : ℝ))
(designatedGroup (d c)) (law c) := by
simp only [projectedReferenceEntropy, Fintype.sum_sigma, groupCap, add_zero,
groupedEntropy]
apply Finset.sum_congr rfl
intro c _
apply Finset.sum_congr rfl
intro k _
rw [baseGroupSize_eq_groupMass]
cases k with
| none =>
change _ * finiteEntropy (residualCenter base d law c) = _
rw [residualCenter_eq_groupLaw]
| some u =>
change _ * finiteEntropy (law c u) = _
rw [weighted_groupLaw_designated_some, groupMass_designated_some]
omit [DecidableEq C] [∀ c, DecidableEq (A c)] in
theorem projectedReferenceEntropy_eq_sum_weighted_groupedEntropy
(base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop)
(law : ∀ c, Shape c → A c → ℝ) :
projectedReferenceEntropy base d (groupCap base d law 0) =
∑ c, (baseSize base c : ℝ) *
groupedEntropy (fun u => (base c u : ℝ) / (baseSize base c : ℝ))
(designatedGroup (d c)) (law c) := by
rw [projectedReferenceEntropy_eq_sum_counts_groupedEntropy]
apply Finset.sum_congr rfl
intro c _
exact groupedEntropy_counts_eq_total_mul (base c) _ _
end MatrixMultiplication.JointProjectedGroupEntropy
end
end OAI
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