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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Population
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.TypeCounts
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointAmbientDegree
open MatrixMultiplication.Foundation JointPopulation
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ)
def localWord (t : JointCoarseHashing.Triple (Position counts)) (h : H) :
Positions counts h → Shape :=
fun i => (t.1 ⟨h, i⟩, t.2.1 ⟨h, i⟩, t.2.2 ⟨h, i⟩)
omit [Fintype H] [DecidableEq H] in
theorem localWord_injective : Function.Injective (localWord counts) := by
intro t u he
apply Prod.ext
· funext p
exact congrArg (fun w => (w p.1 p.2).1) he
· apply Prod.ext
· funext p
exact congrArg (fun w => (w p.1 p.2).2.1) he
· funext p
exact congrArg (fun w => (w p.1 p.2).2.2) he
def localAllowed (sum : H → ℕ) (h : H) (w : Positions counts h → Shape) : Prop :=
(∀ i, (w i).1.val + (w i).2.1.val + (w i).2.2.val = sum h) ∧
∀ s a, wordPopulation (shapeSide s ∘ w) a =
wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a
theorem localAllowed_of_mem_ambient (sum : H → ℕ)
(t : JointCoarseHashing.Triple (Position counts))
(ht : t ∈ ambientSet counts sum) (h : H) :
localAllowed counts sum h (localWord counts t h) := by
obtain ⟨hs, hm⟩ := (Finset.mem_filter.mp ht).2
exact ⟨fun i => hs ⟨h, i⟩, fun s a => hm h s a⟩
def sameSide (s : Fin 3) (t e : JointCoarseHashing.Triple (Position counts)) : Prop :=
∀ h, sideWord counts s t h = sideWord counts s e h
def sharedAmbient (sum : H → ℕ) (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) :
Finset (JointCoarseHashing.Triple (Position counts)) :=
(ambientSet counts sum).filter (sameSide counts s · e)
def classBound (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(h : H) (entropyCap : ℝ) : ℝ :=
((Fintype.card (Positions counts h) : ℝ) + 1) ^ Fintype.card Shape *
Real.exp ((Fintype.card (Positions counts h) : ℝ) *
(entropyCap - finiteEntropy (fun a =>
(wordPopulation (sideWord counts s e h) a : ℝ) /
Fintype.card (Positions counts h))) +
((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) *
(1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ)))
def classExponent (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(h : H) (entropyCap : ℝ) : ℝ :=
(Fintype.card (Positions counts h) : ℝ) *
(entropyCap - finiteEntropy (fun a =>
(wordPopulation (sideWord counts s e h) a : ℝ) /
Fintype.card (Positions counts h))) +
((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) *
(1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ)) +
(Fintype.card Shape : ℝ) * Real.log (Fintype.card (Positions counts h) + 1 : ℕ)
omit [Fintype H] [DecidableEq H] in
theorem classBound_eq_exp (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) (h : H) (entropyCap : ℝ) :
classBound counts s e h entropyCap = Real.exp (classExponent counts s e h entropyCap) := by
unfold classBound classExponent
conv_rhs => rw [Real.exp_add, Real.exp_nat_mul, Real.exp_log (by positivity)]
simp only [Nat.cast_add, Nat.cast_one]
exact mul_comm _ _
omit [Fintype H] [DecidableEq H] in
theorem classExponent_target_eq (s : Fin 3) (e : Target counts) (h : H)
(entropyCap : ℝ) :
classExponent counts s (triple counts e) h entropyCap =
classExponent counts s (triple counts (canonicalTarget counts)) h entropyCap := by
simp_rw [classExponent, marginal_population_eq counts e (canonicalTarget counts)]
theorem sharedAmbient_card_le (sum : H → ℕ) (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h) :
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, classBound counts s e h (entropyCap h) := by
classical
let Allowed := fun h (w : Positions counts h → Shape) =>
localAllowed counts sum h w ∧ ∀ i, shapeSide s (w i) = sideWord counts s e h i
let Local := fun h => {w : Positions counts h → Shape // Allowed h w}
let encode : {t // t ∈ sharedAmbient counts sum s e} → ∀ h, Local h := fun t h =>
⟨localWord counts t.val h, localAllowed_of_mem_ambient counts sum t.val
(Finset.mem_filter.mp t.property).1 h,
fun i => congrFun ((Finset.mem_filter.mp t.property).2 h) i⟩
have hinj : Function.Injective encode := by
intro t u he
apply Subtype.ext
apply localWord_injective counts
funext h
exact congrArg Subtype.val (congrFun he h)
have hcard : (sharedAmbient counts sum s e).card ≤ ∏ h, Fintype.card (Local h) := by
simpa only [Fintype.card_coe, Fintype.card_pi] using
Fintype.card_le_of_injective encode hinj
have hlocal (h : H) : (Fintype.card (Local h) : ℝ) ≤
classBound counts s e h (entropyCap h) := by
exact JointTypeCounts.card_allowed_over_coarse_le (shapeSide s)
(sideWord counts s e h) (Allowed h) (entropyCap h)
(fun w hw => hw.2) (fun w hw => hentropy h w hw.1)
calc
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, (Fintype.card (Local h) : ℝ) := by exact_mod_cast hcard
_ ≤ _ := Finset.prod_le_prod₀ (fun _ _ => Nat.cast_nonneg _) (fun h _ => hlocal h)
omit [Fintype H] [DecidableEq H] in
theorem sharesSide_iff (t e : JointCoarseHashing.Triple (Position counts)) :
JointCoarseHashing.SharesSide t e ↔ ∃ s : Fin 3, sameSide counts s t e := by
constructor
· rintro (hx | hy | hz)
· refine ⟨0, fun h => ?_⟩
funext i
change t.1 ⟨h, i⟩ = e.1 ⟨h, i⟩
exact congrFun hx.symm ⟨h, i⟩
· refine ⟨1, fun h => ?_⟩
funext i
change t.2.1 ⟨h, i⟩ = e.2.1 ⟨h, i⟩
exact congrFun hy.symm ⟨h, i⟩
· refine ⟨2, fun h => ?_⟩
funext i
change t.2.2 ⟨h, i⟩ = e.2.2 ⟨h, i⟩
exact congrFun hz.symm ⟨h, i⟩
· rintro ⟨s, hs⟩
fin_cases s
· apply Or.inl
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
· apply Or.inr ∘ Or.inl
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
· apply Or.inr ∘ Or.inr
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
def neighbors (sum : H → ℕ) (e : JointCoarseHashing.Triple (Position counts)) :
Finset (JointCoarseHashing.Triple (Position counts)) :=
(ambientSet counts sum).filter (fun t => t ≠ e ∧ JointCoarseHashing.SharesSide t e)
theorem neighbors_card_le_sum (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) :
(neighbors counts sum e).card ≤ ∑ s : Fin 3, (sharedAmbient counts sum s e).card := by
classical
have hsub : neighbors counts sum e ⊆
(Finset.univ : Finset (Fin 3)).biUnion (fun s => sharedAmbient counts sum s e) := by
intro t ht
obtain ⟨ha, _, hs⟩ := Finset.mem_filter.mp ht
obtain ⟨s, hs⟩ := (sharesSide_iff counts t e).mp hs
exact Finset.mem_biUnion.mpr
⟨s, Finset.mem_univ _, Finset.mem_filter.mpr ⟨ha, hs⟩⟩
exact (Finset.card_le_card hsub).trans Finset.card_biUnion_le
theorem neighbors_card_le (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h) :
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) := by
calc
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ((sharedAmbient counts sum s e).card : ℝ) := by
exact_mod_cast neighbors_card_le_sum counts sum e
_ ≤ _ := Finset.sum_le_sum fun s _ => sharedAmbient_card_le counts sum s e entropyCap hentropy
theorem neighbors_card_le_three_exp (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h)
(exponent : ℝ)
(hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent) :
((neighbors counts sum e).card : ℝ) ≤ 3 * Real.exp exponent := by
have hprod (s : Fin 3) : (∏ h, classBound counts s e h (entropyCap h)) ≤
Real.exp exponent := by
simp_rw [classBound_eq_exp]
rw [← Real.exp_sum]
exact Real.exp_le_exp.mpr (hupper s)
calc
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) :=
neighbors_card_le counts sum e entropyCap hentropy
_ ≤ ∑ _s : Fin 3, Real.exp exponent := Finset.sum_le_sum fun s _ => hprod s
_ = _ := by simp
theorem log_neighbors_card_le (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h)
(exponent : ℝ)
(hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent)
(hne : 0 < (neighbors counts sum e).card) :
Real.log ((neighbors counts sum e).card : ℝ) ≤ Real.log 3 + exponent := by
have hb := Real.log_le_log (by exact_mod_cast hne :
(0 : ℝ) < (neighbors counts sum e).card)
(neighbors_card_le_three_exp counts sum e entropyCap hentropy exponent hupper)
simpa only [Real.log_mul (by norm_num : (3 : ℝ) ≠ 0) (Real.exp_ne_zero _),
Real.log_exp] using hb
end MatrixMultiplication.JointAmbientDegree
end
end OAI
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