File size: 9,049 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexFiniteEntropy
namespace OAI
/-! Finite entropy, rate estimates and ordered asymptotic limits. -/
namespace MatrixMultiplication.Foundation
open scoped BigOperators
universe u v
noncomputable section
namespace FiniteLaw
theorem map_mass_mul_conditional_map {A B C : Type*}
[Fintype A] [Fintype B] [Fintype C]
(p : FiniteLaw A) (f : A → B) (g : A → C) (b : B) (c : C) :
(p.map f).mass b * ((p.conditional f b).map g).mass c =
(p.map (fun a => (f a, g a))).mass (b, c) := by
classical
rw [map_mass (p.conditional f b) g c, Finset.mul_sum,
map_mass p (fun a => (f a, g a)) (b, c)]
apply Finset.sum_congr rfl
intro a _
by_cases hgc : g a = c
· rw [ite_eq_left hgc, map_mass_mul_conditional]
simp only [Prod.mk.injEq, hgc, and_true]
· rw [ite_eq_right hgc, mul_zero]
have hpair : (f a, g a) ≠ (b, c) :=
fun he => hgc (congrArg Prod.snd he)
rw [ite_eq_right hpair]
end FiniteLaw
def LabelRecord (Label : ℕ → Type u) : ℕ → Type u
| 0 => PUnit
| n + 1 => LabelRecord Label n × Label n
instance labelRecordFintype (Label : ℕ → Type u) [∀ n, Fintype (Label n)] :
(n : ℕ) → Fintype (LabelRecord Label n)
| 0 => by
change Fintype PUnit
infer_instance
| n + 1 => by
letI := labelRecordFintype Label n
change Fintype (LabelRecord Label n × Label n)
infer_instance
def labelRecordOf {A : Type v} {Label : ℕ → Type u}
(labels : ∀ n, A → Label n) : (n : ℕ) → A → LabelRecord Label n
| 0, _ => PUnit.unit
| n + 1, a => (labelRecordOf labels n a, labels n a)
structure FiniteLabelHierarchy (Label : ℕ → Type u) [∀ n, Fintype (Label n)]
(depth : ℕ) where
conditional : ∀ n, n < depth → LabelRecord Label n → FiniteLaw (Label n)
namespace FiniteLabelHierarchy
variable {Label : ℕ → Type u} [∀ n, Fintype (Label n)] {depth : ℕ}
def prefixLaw (h : FiniteLabelHierarchy Label depth) :
(n : ℕ) → n ≤ depth → FiniteLaw (LabelRecord Label n)
| 0, _ =>
{ mass := fun _ => 1
nonneg := fun _ => zero_le_one
total := by
classical
refine Finset.sum_eq_single_of_mem (PUnit.unit : LabelRecord Label 0)
(Finset.mem_univ _) ?_
intro r _ hr
cases r
exact (hr rfl).elim }
| n + 1, hn =>
(h.prefixLaw n (Nat.le_of_succ_le hn)).joint
(h.conditional n (Nat.lt_of_succ_le hn))
@[simp] theorem prefixLaw_zero_mass (h : FiniteLabelHierarchy Label depth)
(hn : 0 ≤ depth) (r : LabelRecord Label 0) :
(h.prefixLaw 0 hn).mass r = 1 := rfl
@[simp] theorem prefixLaw_succ (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n + 1 ≤ depth) :
h.prefixLaw (n + 1) hn =
(h.prefixLaw n (Nat.le_of_succ_le hn)).joint
(h.conditional n (Nat.lt_of_succ_le hn)) := rfl
@[simp] theorem prefixLaw_succ_mass (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n + 1 ≤ depth) (r : LabelRecord Label n) (s : Label n) :
(h.prefixLaw (n + 1) hn).mass (r, s) =
(h.prefixLaw n (Nat.le_of_succ_le hn)).mass r *
(h.conditional n (Nat.lt_of_succ_le hn) r).mass s := rfl
def refinementEntropy (h : FiniteLabelHierarchy Label depth) (n : ℕ) : ℝ :=
if hn : n < depth then
∑ r, (h.prefixLaw n (Nat.le_of_lt hn)).mass r *
finiteEntropy (h.conditional n hn r).mass
else 0
theorem refinementEntropy_eq (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n < depth) :
h.refinementEntropy n =
∑ r, (h.prefixLaw n (Nat.le_of_lt hn)).mass r *
finiteEntropy (h.conditional n hn r).mass := by
simp only [refinementEntropy, dite_eq_left hn]
@[simp] theorem refinementEntropy_eq_zero_of_le
(h : FiniteLabelHierarchy Label depth) {n : ℕ} (hn : depth ≤ n) :
h.refinementEntropy n = 0 := by
simp only [refinementEntropy, dite_eq_right (Nat.not_lt.mpr hn)]
theorem zero_prefix_contribution (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n < depth) (r : LabelRecord Label n)
(hr : (h.prefixLaw n (Nat.le_of_lt hn)).mass r = 0) :
(h.prefixLaw n (Nat.le_of_lt hn)).mass r *
finiteEntropy (h.conditional n hn r).mass = 0 := by
rw [hr, zero_mul]
theorem prefixLaw_entropy_succ (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n + 1 ≤ depth) :
finiteEntropy (h.prefixLaw (n + 1) hn).mass =
finiteEntropy (h.prefixLaw n (Nat.le_of_succ_le hn)).mass +
h.refinementEntropy n := by
rw [refinementEntropy_eq h n (Nat.lt_of_succ_le hn)]
exact FiniteLaw.joint_entropy (h.prefixLaw n (Nat.le_of_succ_le hn))
(h.conditional n (Nat.lt_of_succ_le hn))
theorem prefixLaw_entropy (h : FiniteLabelHierarchy Label depth)
(n : ℕ) (hn : n ≤ depth) :
finiteEntropy (h.prefixLaw n hn).mass =
∑ j ∈ Finset.range n, h.refinementEntropy j := by
induction n with
| zero =>
simp [prefixLaw, finiteEntropy, entropyTerm, LabelRecord]
| succ n ih =>
rw [prefixLaw_entropy_succ h n hn,
ih (Nat.le_of_succ_le hn), Finset.sum_range_succ]
def recordLaw (h : FiniteLabelHierarchy Label depth) :
FiniteLaw (LabelRecord Label depth) := h.prefixLaw depth le_rfl
theorem recordLaw_entropy (h : FiniteLabelHierarchy Label depth) :
finiteEntropy h.recordLaw.mass =
∑ j ∈ Finset.range depth, h.refinementEntropy j :=
h.prefixLaw_entropy depth le_rfl
theorem refinementEntropy_sum_of_injective_record
{A : Type v} [Fintype A] (h : FiniteLabelHierarchy Label depth)
(p : FiniteLaw A) (record : A → LabelRecord Label depth)
(hinjective : Function.Injective record)
(hmass : h.recordLaw.mass = (p.map record).mass) :
(∑ j ∈ Finset.range depth, h.refinementEntropy j) = finiteEntropy p.mass := by
calc
(∑ j ∈ Finset.range depth, h.refinementEntropy j) =
finiteEntropy h.recordLaw.mass := h.recordLaw_entropy.symm
_ = finiteEntropy (p.map record).mass := congrArg finiteEntropy hmass
_ = finiteEntropy p.mass := p.map_entropy_of_injective record hinjective
def ofLawLabels {A : Type v} [Fintype A] (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (depth : ℕ) : FiniteLabelHierarchy Label depth where
conditional n _ r := (p.conditional (labelRecordOf labels n) r).map (labels n)
theorem ofLawLabels_prefixLaw_mass {A : Type v} [Fintype A]
(p : FiniteLaw A) (labels : ∀ n, A → Label n)
(depth n : ℕ) (hn : n ≤ depth) :
((ofLawLabels p labels depth).prefixLaw n hn).mass =
(p.map (labelRecordOf labels n)).mass := by
classical
induction n with
| zero =>
funext r
rw [prefixLaw_zero_mass, FiniteLaw.map_mass]
refine p.total.symm.trans ?_
apply Finset.sum_congr rfl
intro a _
rw [ite_eq_left (show labelRecordOf labels 0 a = r by cases r; rfl)]
| succ n ih =>
funext rs
rcases rs with ⟨r, s⟩
rw [prefixLaw_succ_mass]
change ((ofLawLabels p labels depth).prefixLaw n
(Nat.le_of_succ_le hn)).mass r *
((p.conditional (labelRecordOf labels n) r).map (labels n)).mass s =
(p.map (labelRecordOf labels (n + 1))).mass (r, s)
rw [ih (Nat.le_of_succ_le hn)]
exact p.map_mass_mul_conditional_map
(labelRecordOf labels n) (labels n) r s
theorem ofLawLabels_refinementEntropy_eq {A : Type v} [Fintype A]
(p : FiniteLaw A) (labels : ∀ n, A → Label n)
(depth n : ℕ) (hn : n < depth) :
(ofLawLabels p labels depth).refinementEntropy n =
∑ r : LabelRecord Label n, (p.map (labelRecordOf labels n)).mass r *
finiteEntropy ((p.conditional (labelRecordOf labels n) r).map (labels n)).mass := by
rw [refinementEntropy_eq _ n hn,
ofLawLabels_prefixLaw_mass p labels depth n (Nat.le_of_lt hn)]
rfl
theorem ofLawLabels_refinementEntropy_sum {A : Type v} [Fintype A]
(p : FiniteLaw A) (labels : ∀ n, A → Label n) (depth : ℕ)
(hinjective : Function.Injective (labelRecordOf labels depth)) :
(∑ j ∈ Finset.range depth,
(ofLawLabels p labels depth).refinementEntropy j) = finiteEntropy p.mass := by
apply (ofLawLabels p labels depth).refinementEntropy_sum_of_injective_record
p (labelRecordOf labels depth) hinjective
exact ofLawLabels_prefixLaw_mass p labels depth depth le_rfl
theorem entropy_eq_sum_conditional_labels {A : Type v} [Fintype A]
(p : FiniteLaw A) (labels : ∀ n, A → Label n) (depth : ℕ)
(hinjective : Function.Injective (labelRecordOf labels depth)) :
finiteEntropy p.mass =
∑ j ∈ Finset.range depth, ∑ r : LabelRecord Label j,
(p.map (labelRecordOf labels j)).mass r *
finiteEntropy ((p.conditional (labelRecordOf labels j) r).map (labels j)).mass := by
rw [← ofLawLabels_refinementEntropy_sum p labels depth hinjective]
apply Finset.sum_congr rfl
intro j hj
exact ofLawLabels_refinementEntropy_eq p labels depth j (Finset.mem_range.mp hj)
end FiniteLabelHierarchy
end
end MatrixMultiplication.Foundation
end OAI
|