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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AmbientDegree
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.GibbsApproximation
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedStatisticWindows
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
open scoped BigOperators Topology
open Filter MatrixMultiplication.Foundation
open MatrixMultiplication.JointPopulation MatrixMultiplication.JointAmbientDegree
open MatrixMultiplication.JointGibbsApproximation
namespace MatrixMultiplication.JointAmbientEntropy
section Empirical
variable {I A B : Type*} [Fintype I] [DecidableEq I]
[Fintype A] [DecidableEq A] [Fintype B] [DecidableEq B]
def empiricalFiniteLaw (w : I → A) (hpos : 0 < Fintype.card I) : FiniteLaw A where
mass a := (wordPopulation w a : ℝ) / Fintype.card I
nonneg a := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
total := by
rw [← Finset.sum_div, ← Nat.cast_sum, wordPopulation_sum]
exact div_self (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hpos))
omit [DecidableEq I] in
@[simp] theorem empiricalFiniteLaw_mass (w : I → A) (hpos : 0 < Fintype.card I) (a : A) :
(empiricalFiniteLaw w hpos).mass a =
(wordPopulation w a : ℝ) / Fintype.card I := rfl
omit [DecidableEq I] in
theorem empiricalFiniteLaw_map_mass (w : I → A) (hpos : 0 < Fintype.card I)
(f : A → B) (b : B) :
((empiricalFiniteLaw w hpos).map f).mass b =
(wordPopulation (f ∘ w) b : ℝ) / Fintype.card I := by
rw [FiniteLaw.map_mass, InheritedMasks.wordPopulation_statistic,
Nat.cast_sum, Finset.sum_div, Finset.sum_filter]
simp only [empiricalFiniteLaw_mass]
omit [DecidableEq I] in
theorem empiricalFiniteLaw_supported (w : I → A) (hpos : 0 < Fintype.card I)
(p : FiniteLaw A) (hsupport : ∀ i, 0 < p.mass (w i)) :
∀ a, p.mass a = 0 → (empiricalFiniteLaw w hpos).mass a = 0 := by
intro a ha
have : IsEmpty {i : I // w i = a} := ⟨fun i => by
have hp := hsupport i.val
rw [i.property, ha] at hp
exact lt_irrefl 0 hp⟩
simp only [empiricalFiniteLaw_mass, wordPopulation, Fintype.card_of_isEmpty,
Nat.cast_zero, zero_div]
end Empirical
variable {H : Type*} [Fintype H] [DecidableEq H]
def canonicalLaw (counts : H → Shape → ℕ) (h : H) : FiniteLaw Shape :=
if hpos : 0 < Fintype.card (Positions counts h) then
empiricalFiniteLaw ((canonicalTarget counts) h).val hpos
else FiniteLaw.pure (0, 0, 0)
omit [Fintype H] [DecidableEq H] in
theorem canonicalLaw_mass_of_pos (counts : H → Shape → ℕ) (h : H)
(hpos : 0 < Fintype.card (Positions counts h)) (a : Shape) :
(canonicalLaw counts h).mass a =
(counts h a : ℝ) / Fintype.card (Positions counts h) := by
rw [canonicalLaw, dite_eq_left hpos, empiricalFiniteLaw_mass,
((canonicalTarget counts) h).property]
omit [Fintype H] [DecidableEq H] in
theorem total_mul_canonicalLaw_mass (counts : H → Shape → ℕ) (h : H) (a : Shape) :
(Fintype.card (Positions counts h) : ℝ) * (canonicalLaw counts h).mass a =
counts h a := by
by_cases hpos : 0 < Fintype.card (Positions counts h)
· rw [canonicalLaw_mass_of_pos counts h hpos]
exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hpos))
· have hzero : Fintype.card (Positions counts h) = 0 := Nat.eq_zero_of_not_pos hpos
have hsum : ∑ a, counts h a = 0 := by simpa only [Positions, Fintype.card_fin] using hzero
have ha : counts h a = 0 := by
have hle : counts h a ≤ ∑ b, counts h b :=
Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ a)
rw [hsum] at hle
exact Nat.eq_zero_of_le_zero hle
simp only [hzero, ha, Nat.cast_zero, zero_mul]
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_marginal_eq (counts : H → Shape → ℕ) (sum : H → ℕ)
(h : H) (w : Positions counts h → Shape)
(hpos : 0 < Fintype.card (Positions counts h))
(hw : localAllowed counts sum h w) (s : Fin 3) :
((empiricalFiniteLaw w hpos).map (shapeSide s)).mass =
((canonicalLaw counts h).map (shapeSide s)).mass := by
rw [canonicalLaw, dite_eq_left hpos]
funext a
rw [empiricalFiniteLaw_map_mass, empiricalFiniteLaw_map_mass]
exact congrArg (fun n : ℕ => (n : ℝ) / Fintype.card (Positions counts h)) (hw.2 s a)
def gibbsCap (counts : H → Shape → ℕ) (h : H) (p : FiniteLaw Shape)
(u v w : Fin 17 → ℝ) : ℝ :=
finiteEntropy p.mass + marginalError p (canonicalLaw counts h) (shapeSide 0) u +
marginalError p (canonicalLaw counts h) (shapeSide 1) v +
marginalError p (canonicalLaw counts h) (shapeSide 2) w
omit [Fintype H] [DecidableEq H] in
theorem gibbsCap_nonneg (counts : H → Shape → ℕ) (h : H) (p : FiniteLaw Shape)
(u v w : Fin 17 → ℝ) : 0 ≤ gibbsCap counts h p u v w := by
exact add_nonneg
(add_nonneg (add_nonneg p.entropy_nonneg (marginalError_nonneg _ _ _ _))
(marginalError_nonneg _ _ _ _)) (marginalError_nonneg _ _ _ _)
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_entropy_le_of_word_support (counts : H → Shape → ℕ) (sum : H → ℕ)
(h : H) (w : Positions counts h → Shape) (p : FiniteLaw Shape)
(u v z : Fin 17 → ℝ) (c : ℝ)
(hsupport : ∀ i, 0 < p.mass (w i))
(hlog : ∀ a, 0 < p.mass a → Real.log (p.mass a) =
u (shapeSide 0 a) + v (shapeSide 1 a) + z (shapeSide 2 a) - c)
(hw : localAllowed counts sum h w) :
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ gibbsCap counts h p u v z := by
by_cases hpos : 0 < Fintype.card (Positions counts h)
· have hs := empiricalFiniteLaw_supported w hpos p hsupport
have hg := entropy_le_of_marginal_errors_of_support p (empiricalFiniteLaw w hpos)
(shapeSide 0) (shapeSide 1) (shapeSide 2) u v z c hs hlog
have hx := localAllowed_marginal_eq counts sum h w hpos hw (0 : Fin 3)
have hy := localAllowed_marginal_eq counts sum h w hpos hw (1 : Fin 3)
have hz := localAllowed_marginal_eq counts sum h w hpos hw (2 : Fin 3)
have hmass : (empiricalFiniteLaw w hpos).mass =
(fun a => (wordPopulation w a : ℝ) / Fintype.card (Positions counts h)) := by
funext a
exact empiricalFiniteLaw_mass w hpos a
simpa only [gibbsCap, marginalError, empiricalFiniteLaw_mass, hmass, hx, hy, hz] using hg
· have hzero : Fintype.card (Positions counts h) = 0 := Nat.eq_zero_of_not_pos hpos
simpa only [hzero, Nat.cast_zero, div_zero, finiteEntropy, entropyTerm_zero,
Finset.sum_const_zero] using gibbsCap_nonneg counts h p u v z
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_entropy_le (counts : H → Shape → ℕ) (sum : H → ℕ)
(h : H) (w : Positions counts h → Shape) (p : FiniteLaw Shape)
(u v z : Fin 17 → ℝ) (c : ℝ)
(hsupport : ∀ a : Shape, a.1.val + a.2.1.val + a.2.2.val = sum h → 0 < p.mass a)
(hlog : ∀ a, 0 < p.mass a → Real.log (p.mass a) =
u (shapeSide 0 a) + v (shapeSide 1 a) + z (shapeSide 2 a) - c)
(hw : localAllowed counts sum h w) :
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ gibbsCap counts h p u v z :=
localAllowed_entropy_le_of_word_support counts sum h w p u v z c
(fun i => hsupport (w i) (hw.1 i)) hlog hw
theorem sharedAmbient_card_le_of_supported_gibbs (counts : H → Shape → ℕ) (sum : H → ℕ)
(s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(p : H → FiniteLaw Shape) (u v w : H → Fin 17 → ℝ) (c : H → ℝ)
(hsupport : ∀ h word, localAllowed counts sum h word → ∀ i, 0 < (p h).mass (word i))
(hlog : ∀ h a, 0 < (p h).mass a → Real.log ((p h).mass a) =
u h (shapeSide 0 a) + v h (shapeSide 1 a) + w h (shapeSide 2 a) - c h) :
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, classBound counts s e h (gibbsCap counts h (p h) (u h) (v h) (w h)) :=
sharedAmbient_card_le counts sum s e _
(fun h word hw => localAllowed_entropy_le_of_word_support counts sum h word (p h)
(u h) (v h) (w h) (c h) (hsupport h word hw) (hlog h) hw)
theorem neighbors_card_le_of_supported_gibbs (counts : H → Shape → ℕ) (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts))
(p : H → FiniteLaw Shape) (u v w : H → Fin 17 → ℝ) (c : H → ℝ)
(hsupport : ∀ h word, localAllowed counts sum h word → ∀ i, 0 < (p h).mass (word i))
(hlog : ∀ h a, 0 < (p h).mass a → Real.log ((p h).mass a) =
u h (shapeSide 0 a) + v h (shapeSide 1 a) + w h (shapeSide 2 a) - c h) :
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h
(gibbsCap counts h (p h) (u h) (v h) (w h)) :=
neighbors_card_le counts sum e _
(fun h word hw => localAllowed_entropy_le_of_word_support counts sum h word (p h)
(u h) (v h) (w h) (c h) (hsupport h word hw) (hlog h) hw)
theorem sharedAmbient_card_le_of_gibbs (counts : H → Shape → ℕ) (sum : H → ℕ)
(s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(p : H → FiniteLaw Shape) (u v w : H → Fin 17 → ℝ) (c : H → ℝ)
(hsupport : ∀ h (a : Shape), a.1.val + a.2.1.val + a.2.2.val = sum h → 0 < (p h).mass a)
(hlog : ∀ h a, 0 < (p h).mass a → Real.log ((p h).mass a) =
u h (shapeSide 0 a) + v h (shapeSide 1 a) + w h (shapeSide 2 a) - c h) :
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, classBound counts s e h (gibbsCap counts h (p h) (u h) (v h) (w h)) :=
sharedAmbient_card_le counts sum s e _
(fun h word hw => localAllowed_entropy_le counts sum h word (p h)
(u h) (v h) (w h) (c h) (hsupport h) (hlog h) hw)
theorem neighbors_card_le_of_gibbs (counts : H → Shape → ℕ) (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts))
(p : H → FiniteLaw Shape) (u v w : H → Fin 17 → ℝ) (c : H → ℝ)
(hsupport : ∀ h (a : Shape), a.1.val + a.2.1.val + a.2.2.val = sum h → 0 < (p h).mass a)
(hlog : ∀ h a, 0 < (p h).mass a → Real.log ((p h).mass a) =
u h (shapeSide 0 a) + v h (shapeSide 1 a) + w h (shapeSide 2 a) - c h) :
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h
(gibbsCap counts h (p h) (u h) (v h) (w h)) :=
neighbors_card_le counts sum e _
(fun h word hw => localAllowed_entropy_le counts sum h word (p h)
(u h) (v h) (w h) (c h) (hsupport h) (hlog h) hw)
def classScale (counts : H → Shape → ℕ) (h : H) (N : ℕ) : ℝ :=
(Fintype.card (Positions counts h) : ℝ) / N
omit [Fintype H] [DecidableEq H] in
theorem classScale_nonneg (counts : H → Shape → ℕ) (h : H) (N : ℕ) :
0 ≤ classScale counts h N := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
omit [Fintype H] [DecidableEq H] in
theorem classScale_mul_canonicalLaw_mass (counts : H → Shape → ℕ) (h : H)
(N : ℕ) (a : Shape) :
classScale counts h N * (canonicalLaw counts h).mass a = (counts h a : ℝ) / N := by
unfold classScale
calc
_ = ((Fintype.card (Positions counts h) : ℝ) * (canonicalLaw counts h).mass a) / N := by ring
_ = _ := by rw [total_mul_canonicalLaw_mass]
omit [Fintype H] [DecidableEq H] in
theorem classScale_mul_canonicalLaw_map_mass {B : Type*} [Fintype B] [DecidableEq B]
(counts : H → Shape → ℕ) (h : H) (N : ℕ) (f : Shape → B) (b : B) :
classScale counts h N * ((canonicalLaw counts h).map f).mass b =
∑ a, if f a = b then (counts h a : ℝ) / N else 0 := by
rw [FiniteLaw.map_mass, Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
by_cases ha : f a = b
· simp only [ha, ite_true]
exact classScale_mul_canonicalLaw_mass counts h N a
· simp only [ha, ite_false, mul_zero]
omit [Fintype H] [DecidableEq H] in
theorem classScale_mul_marginalError {B : Type*} [Fintype B] [DecidableEq B]
(counts : H → Shape → ℕ) (h : H) (N : ℕ) (p : FiniteLaw Shape)
(f : Shape → B) (v : B → ℝ) :
classScale counts h N * marginalError p (canonicalLaw counts h) f v =
∑ b, |(∑ a, if f a = b then (counts h a : ℝ) / N else 0) -
classScale counts h N * (p.map f).mass b| * |v b| := by
unfold marginalError
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro b _
have habs : classScale counts h N *
|((canonicalLaw counts h).map f).mass b - (p.map f).mass b| =
|classScale counts h N * ((canonicalLaw counts h).map f).mass b -
classScale counts h N * (p.map f).mass b| := by
rw [← mul_sub, abs_mul, abs_of_nonneg (classScale_nonneg counts h N)]
rw [← mul_assoc, habs, classScale_mul_canonicalLaw_map_mass]
omit [Fintype H] [DecidableEq H] in
theorem tendsto_classScale (counts : ℕ → H → Shape → ℕ) (h : H)
(p : FiniteLaw Shape) (mass : ℝ)
(hcounts : ∀ a, Tendsto (fun N => (counts N h a : ℝ) / N) atTop (𝓝 (mass * p.mass a))) :
Tendsto (fun N => classScale (counts N) h N) atTop (𝓝 mass) := by
have hsum := tendsto_finsetSum Finset.univ (fun a _ => hcounts a)
simpa only [classScale, Positions, Fintype.card_fin, Nat.cast_sum,
Finset.sum_div, ← Finset.mul_sum, p.total, mul_one] using hsum
omit [Fintype H] [DecidableEq H] in
theorem tendsto_classScale_mul_marginalError {B : Type*} [Fintype B] [DecidableEq B]
(counts : ℕ → H → Shape → ℕ) (h : H) (p : FiniteLaw Shape)
(mass : ℝ) (f : Shape → B) (v : B → ℝ)
(hcounts : ∀ a, Tendsto (fun N => (counts N h a : ℝ) / N) atTop (𝓝 (mass * p.mass a))) :
Tendsto (fun N => classScale (counts N) h N *
marginalError p (canonicalLaw (counts N) h) f v) atTop (𝓝 0) := by
have htotal := tendsto_classScale counts h p mass hcounts
have hside (b : B) :
Tendsto (fun N => ∑ a, if f a = b then (counts N h a : ℝ) / N else 0)
atTop (𝓝 (mass * (p.map f).mass b)) := by
have hterm (a : Shape) :
Tendsto (fun N => if f a = b then (counts N h a : ℝ) / N else 0)
atTop (𝓝 (if f a = b then mass * p.mass a else 0)) := by
by_cases ha : f a = b
· simpa only [ha, ite_true] using hcounts a
· simp only [ha, ite_false]
exact tendsto_const_nhds
simpa only [FiniteLaw.map_mass, Finset.mul_sum, mul_ite, mul_zero] using
tendsto_finsetSum Finset.univ (fun a _ => hterm a)
have hterm (b : B) :
Tendsto (fun N => |(∑ a, if f a = b then (counts N h a : ℝ) / N else 0) -
classScale (counts N) h N * (p.map f).mass b| * |v b|) atTop (𝓝 0) := by
simpa only [sub_self, abs_zero, zero_mul] using
((hside b).sub (htotal.mul_const ((p.map f).mass b))).abs.mul_const |v b|
simpa only [classScale_mul_marginalError, Finset.sum_const_zero] using
tendsto_finsetSum Finset.univ (fun b _ => hterm b)
omit [Fintype H] [DecidableEq H] in
theorem tendsto_classScale_mul_gibbsCap (counts : ℕ → H → Shape → ℕ) (h : H)
(p : FiniteLaw Shape) (mass : ℝ) (u v w : Fin 17 → ℝ)
(hcounts : ∀ a, Tendsto (fun N => (counts N h a : ℝ) / N) atTop (𝓝 (mass * p.mass a))) :
Tendsto (fun N => classScale (counts N) h N *
gibbsCap (counts N) h p u v w) atTop (𝓝 (mass * finiteEntropy p.mass)) := by
have hm := (tendsto_classScale counts h p mass hcounts).mul_const (finiteEntropy p.mass)
have hx := tendsto_classScale_mul_marginalError counts h p mass (shapeSide 0) u hcounts
have hy := tendsto_classScale_mul_marginalError counts h p mass (shapeSide 1) v hcounts
have hz := tendsto_classScale_mul_marginalError counts h p mass (shapeSide 2) w hcounts
simpa only [gibbsCap, mul_add, add_zero] using ((hm.add hx).add hy).add hz
end MatrixMultiplication.JointAmbientEntropy
end
end OAI
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