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cdec422 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.GroupedEntropyPartition
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointGroupedEntropyEmbedding
open MatrixMultiplication.Foundation JointCompatibilityIncidence JointNativeGroupedEntropy
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {U V K L A : Type*}
[Fintype U] [Fintype V] [Fintype K] [Fintype L] [Fintype A]
omit [Fintype K] [Fintype L] in
theorem groupMass_label_injective (j : K → L) (hj : Function.Injective j)
(p : U → ℝ) (group : U → K) (k : K) :
groupMass p (j ∘ group) (j k) = groupMass p group k := by
simp only [groupMass, Function.comp_apply, hj.eq_iff]
omit [Fintype K] [Fintype L] [Fintype A] in
theorem groupLaw_label_injective (j : K → L) (hj : Function.Injective j)
(p : U → ℝ) (group : U → K) (q : U → A → ℝ) (k : K) :
groupLaw p (j ∘ group) q (j k) = groupLaw p group q k := by
funext a
simp only [groupLaw, groupMass_label_injective j hj,
Function.comp_apply, hj.eq_iff]
omit [Fintype K] [Fintype L] in
theorem groupMass_label_not_mem_range (j : K → L) (p : U → ℝ)
(group : U → K) (l : L) (hl : l ∉ Set.range j) :
groupMass p (j ∘ group) l = 0 := by
apply Finset.sum_eq_zero
intro u _
exact ite_eq_right (fun equality => hl ⟨group u, equality⟩)
theorem groupedEntropy_label_injective (j : K → L) (hj : Function.Injective j)
(p : U → ℝ) (group : U → K) (q : U → A → ℝ) :
groupedEntropy p (j ∘ group) q = groupedEntropy p group q := by
unfold groupedEntropy
symm
apply Fintype.sum_of_injective j hj
· intro l hl
rw [groupMass_label_not_mem_range j p group l hl, zero_mul]
· intro k
rw [groupMass_label_injective j hj, groupLaw_label_injective j hj]
def pushforwardWeight (p : U → ℝ) (f : U → V) (v : V) : ℝ :=
∑ u, if f u = v then p u else 0
theorem sum_pushforwardWeight_mul (p : U → ℝ) (f : U → V) (test : V → ℝ) :
(∑ v, pushforwardWeight p f v * test v) = ∑ u, p u * test (f u) := by
simp only [pushforwardWeight, Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro u _
simp only [ite_mul, zero_mul]
simp
omit [Fintype K] in
theorem groupMass_pushforward (p : U → ℝ) (f : U → V)
(group : V → K) (k : K) :
groupMass (pushforwardWeight p f) group k = groupMass p (group ∘ f) k := by
simpa only [groupMass, mul_ite, mul_one, mul_zero, Function.comp_apply] using
(sum_pushforwardWeight_mul p f (fun v => if group v = k then 1 else 0))
omit [Fintype K] [Fintype A] in
theorem groupMeasure_pushforward (p : U → ℝ) (f : U → V)
(group : V → K) (q : V → A → ℝ) (k : K) :
groupMeasure (pushforwardWeight p f) group q k =
groupMeasure p (group ∘ f) (q ∘ f) k := by
funext a
simpa only [groupMeasure, mul_ite, mul_zero, Function.comp_apply] using
(sum_pushforwardWeight_mul p f (fun v => if group v = k then q v a else 0))
omit [Fintype K] [Fintype A] in
theorem groupLaw_pushforward (p : U → ℝ) (f : U → V)
(group : V → K) (q : V → A → ℝ) (k : K) :
groupLaw (pushforwardWeight p f) group q k =
groupLaw p (group ∘ f) (q ∘ f) k := by
funext a
change groupMeasure (pushforwardWeight p f) group q k a /
groupMass (pushforwardWeight p f) group k =
groupMeasure p (group ∘ f) (q ∘ f) k a / groupMass p (group ∘ f) k
rw [groupMeasure_pushforward, groupMass_pushforward]
theorem groupedEntropy_pushforward (p : U → ℝ) (f : U → V)
(group : V → K) (q : V → A → ℝ) :
groupedEntropy (pushforwardWeight p f) group q =
groupedEntropy p (group ∘ f) (q ∘ f) := by
simp only [groupedEntropy, groupMass_pushforward, groupLaw_pushforward]
theorem groupedEntropy_pushforward_of_agree (p : U → ℝ) (f : U → V)
(group : V → K) (qU : U → A → ℝ) (qV : V → A → ℝ)
(hq : ∀ u, qV (f u) = qU u) :
groupedEntropy (pushforwardWeight p f) group qV =
groupedEntropy p (group ∘ f) qU := by
rw [groupedEntropy_pushforward]
have he : qV ∘ f = qU := funext hq
rw [he]
theorem groupedEntropy_pushforward_embedding (p : U → ℝ) (f : U → V)
(groupU : U → K) (groupV : V → L) (j : K → L) (hj : Function.Injective j)
(hgroup : ∀ u, groupV (f u) = j (groupU u))
(qU : U → A → ℝ) (qV : V → A → ℝ) (hq : ∀ u, qV (f u) = qU u) :
groupedEntropy (pushforwardWeight p f) groupV qV = groupedEntropy p groupU qU := by
rw [groupedEntropy_pushforward_of_agree p f groupV qU qV hq]
have hg : groupV ∘ f = j ∘ groupU := funext hgroup
rw [hg, groupedEntropy_label_injective j hj]
theorem sum_map_mass_mul (p : FiniteLaw U) (f : U → V) (test : V → ℝ) :
(∑ v, (p.map f).mass v * test v) = ∑ u, p.mass u * test (f u) := by
simpa only [FiniteLaw.map_mass, pushforwardWeight] using
(sum_pushforwardWeight_mul p.mass f test)
omit [Fintype K] in
theorem groupMass_map (p : FiniteLaw U) (f : U → V)
(group : V → K) (k : K) :
groupMass (p.map f).mass group k = groupMass p.mass (group ∘ f) k := by
simpa only [groupMass, mul_ite, mul_one, mul_zero, Function.comp_apply] using
(sum_map_mass_mul p f (fun v => if group v = k then 1 else 0))
omit [Fintype K] [Fintype A] in
theorem groupMeasure_map (p : FiniteLaw U) (f : U → V)
(group : V → K) (q : V → A → ℝ) (k : K) :
groupMeasure (p.map f).mass group q k =
groupMeasure p.mass (group ∘ f) (q ∘ f) k := by
funext a
simpa only [groupMeasure, mul_ite, mul_zero, Function.comp_apply] using
(sum_map_mass_mul p f (fun v => if group v = k then q v a else 0))
omit [Fintype K] [Fintype A] in
theorem groupLaw_map (p : FiniteLaw U) (f : U → V)
(group : V → K) (q : V → A → ℝ) (k : K) :
groupLaw (p.map f).mass group q k = groupLaw p.mass (group ∘ f) (q ∘ f) k := by
funext a
change groupMeasure (p.map f).mass group q k a / groupMass (p.map f).mass group k =
groupMeasure p.mass (group ∘ f) (q ∘ f) k a / groupMass p.mass (group ∘ f) k
rw [groupMeasure_map, groupMass_map]
theorem groupedEntropy_map (p : FiniteLaw U) (f : U → V)
(group : V → K) (q : V → A → ℝ) :
groupedEntropy (p.map f).mass group q =
groupedEntropy p.mass (group ∘ f) (q ∘ f) := by
simp only [groupedEntropy, groupMass_map, groupLaw_map]
theorem groupedEntropy_map_of_agree (p : FiniteLaw U) (f : U → V)
(group : V → K) (qU : U → A → ℝ) (qV : V → A → ℝ)
(hq : ∀ u, qV (f u) = qU u) :
groupedEntropy (p.map f).mass group qV =
groupedEntropy p.mass (group ∘ f) qU := by
rw [groupedEntropy_map]
have he : qV ∘ f = qU := funext hq
rw [he]
theorem groupedEntropy_map_embedding (p : FiniteLaw U) (f : U → V)
(groupU : U → K) (groupV : V → L) (j : K → L) (hj : Function.Injective j)
(hgroup : ∀ u, groupV (f u) = j (groupU u))
(qU : U → A → ℝ) (qV : V → A → ℝ) (hq : ∀ u, qV (f u) = qU u) :
groupedEntropy (p.map f).mass groupV qV = groupedEntropy p.mass groupU qU := by
rw [groupedEntropy_map_of_agree p f groupV qU qV hq]
have hg : groupV ∘ f = j ∘ groupU := funext hgroup
rw [hg, groupedEntropy_label_injective j hj]
end MatrixMultiplication.JointGroupedEntropyEmbedding
end
end OAI
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