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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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity
import Mathlib.Tactic.Linarith
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
open scoped BigOperators
open MatrixMultiplication.Foundation
namespace MatrixMultiplication.JointEntropyMax
theorem entropy_nonneg {A : Type*} [Fintype A] (p : FiniteLaw A) :
0 ≤ finiteEntropy p.mass :=
FiniteLaw.entropy_nonneg p
theorem map_entropy_le {A B : Type*} [Fintype A] [Fintype B]
(p : FiniteLaw A) (f : A → B) :
finiteEntropy (p.map f).mass ≤ finiteEntropy p.mass :=
FiniteLaw.map_entropy_le p f
theorem entropy_sub_map_nonneg {A B : Type*} [Fintype A] [Fintype B]
(p : FiniteLaw A) (f : A → B) :
0 ≤ finiteEntropy p.mass - finiteEntropy (p.map f).mass :=
sub_nonneg.mpr (map_entropy_le p f)
theorem entropyTerm_le_crossEntropyTerm {p q : ℝ} (hp : 0 < p) (hq : 0 ≤ q) :
entropyTerm q ≤ -q * Real.log p + p - q := by
by_cases hzero : q = 0
· simpa [hzero] using hp.le
have hqpos : 0 < q := lt_of_le_of_ne hq (Ne.symm hzero)
have hlog := Real.log_le_sub_one_of_pos (div_pos hp hqpos)
rw [Real.log_div hp.ne' hqpos.ne'] at hlog
have hmul := mul_le_mul_of_nonneg_left hlog hq
have hcancel : q * (p / q - 1) = p - q := by
field_simp
rw [hcancel] at hmul
unfold entropyTerm
linarith
theorem entropy_le_crossEntropy {A : Type*} [Fintype A]
(p q : FiniteLaw A) (hp : ∀ a, 0 < p.mass a) :
finiteEntropy q.mass ≤ ∑ a, -q.mass a * Real.log (p.mass a) := by
calc
finiteEntropy q.mass ≤
∑ a, (-q.mass a * Real.log (p.mass a) + p.mass a - q.mass a) :=
Finset.sum_le_sum fun a _ => entropyTerm_le_crossEntropyTerm (hp a) (q.nonneg a)
_ = ∑ a, -q.mass a * Real.log (p.mass a) := by
rw [Finset.sum_sub_distrib, Finset.sum_add_distrib, p.total, q.total]
ring
theorem sum_mass_mul_coordinate {A B : Type*} [Fintype A] [Fintype B]
(p : FiniteLaw A) (f : A → B) (v : B → ℝ) :
(∑ a, p.mass a * v (f a)) = ∑ b, (p.map f).mass b * v b := by
classical
simp_rw [FiniteLaw.map_mass, Finset.sum_mul, ite_mul, zero_mul]
rw [Finset.sum_comm]
simp
theorem sum_mass_mul_coordinate_eq {A B : Type*} [Fintype A] [Fintype B]
(p q : FiniteLaw A) (f : A → B) (v : B → ℝ)
(hmarginal : (q.map f).mass = (p.map f).mass) :
(∑ a, q.mass a * v (f a)) = ∑ a, p.mass a * v (f a) := by
rw [sum_mass_mul_coordinate q f v, sum_mass_mul_coordinate p f v, hmarginal]
theorem entropy_le_of_three_marginals
{A X Y Z : Type*} [Fintype A] [Fintype X] [Fintype Y] [Fintype Z]
(p q : FiniteLaw A) (fX : A → X) (fY : A → Y) (fZ : A → Z)
(u : X → ℝ) (v : Y → ℝ) (w : Z → ℝ) (c : ℝ)
(hp : ∀ a, 0 < p.mass a)
(hlog : ∀ a, Real.log (p.mass a) = u (fX a) + v (fY a) + w (fZ a) - c)
(hX : (q.map fX).mass = (p.map fX).mass)
(hY : (q.map fY).mass = (p.map fY).mass)
(hZ : (q.map fZ).mass = (p.map fZ).mass) :
finiteEntropy q.mass ≤ finiteEntropy p.mass := by
have hx := sum_mass_mul_coordinate_eq p q fX u hX
have hy := sum_mass_mul_coordinate_eq p q fY v hY
have hz := sum_mass_mul_coordinate_eq p q fZ w hZ
have hexpand (r : FiniteLaw A) :
(∑ a, r.mass a * (u (fX a) + v (fY a) + w (fZ a) - c)) =
(∑ a, r.mass a * u (fX a)) + (∑ a, r.mass a * v (fY a)) +
(∑ a, r.mass a * w (fZ a)) - c := by
simp_rw [mul_sub, mul_add]
rw [Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_add_distrib,
← Finset.sum_mul, r.total, one_mul]
have hexpect : (∑ a, q.mass a * Real.log (p.mass a)) =
∑ a, p.mass a * Real.log (p.mass a) := by
simp_rw [hlog]
rw [hexpand q, hexpand p, hx, hy, hz]
calc
finiteEntropy q.mass ≤ ∑ a, -q.mass a * Real.log (p.mass a) :=
entropy_le_crossEntropy p q hp
_ = -(∑ a, q.mass a * Real.log (p.mass a)) := by
simp only [neg_mul, Finset.sum_neg_distrib]
_ = -(∑ a, p.mass a * Real.log (p.mass a)) := by rw [hexpect]
_ = finiteEntropy p.mass := by
simp only [finiteEntropy, entropyTerm, neg_mul, Finset.sum_neg_distrib]
def normalizedPositiveLaw {A : Type*} [Fintype A] [Nonempty A]
(weight : A → ℝ) (hweight : ∀ a, 0 < weight a) : FiniteLaw A where
mass a := weight a / ∑ b, weight b
nonneg a := div_nonneg (hweight a).le
(Finset.sum_nonneg fun b _ => (hweight b).le)
total := by
have htotal : 0 < ∑ b, weight b :=
Finset.sum_pos (fun b _ => hweight b) Finset.univ_nonempty
rw [← Finset.sum_div, div_self htotal.ne']
theorem normalizedPositiveLaw_mass_pos {A : Type*} [Fintype A] [Nonempty A]
(weight : A → ℝ) (hweight : ∀ a, 0 < weight a) (a : A) :
0 < (normalizedPositiveLaw weight hweight).mass a := by
exact div_pos (hweight a)
(Finset.sum_pos (fun b _ => hweight b) Finset.univ_nonempty)
theorem entropy_le_normalized_product
{A X Y Z : Type*} [Fintype A] [Nonempty A]
[Fintype X] [Fintype Y] [Fintype Z]
(fX : A → X) (fY : A → Y) (fZ : A → Z)
(wx : X → ℝ) (wy : Y → ℝ) (wz : Z → ℝ)
(hx : ∀ x, 0 < wx x) (hy : ∀ y, 0 < wy y) (hz : ∀ z, 0 < wz z)
(q : FiniteLaw A)
(hX : (q.map fX).mass =
((normalizedPositiveLaw (fun a => wx (fX a) * wy (fY a) * wz (fZ a))
(fun _ => mul_pos (mul_pos (hx _) (hy _)) (hz _))).map fX).mass)
(hY : (q.map fY).mass =
((normalizedPositiveLaw (fun a => wx (fX a) * wy (fY a) * wz (fZ a))
(fun _ => mul_pos (mul_pos (hx _) (hy _)) (hz _))).map fY).mass)
(hZ : (q.map fZ).mass =
((normalizedPositiveLaw (fun a => wx (fX a) * wy (fY a) * wz (fZ a))
(fun _ => mul_pos (mul_pos (hx _) (hy _)) (hz _))).map fZ).mass) :
finiteEntropy q.mass ≤
finiteEntropy (normalizedPositiveLaw
(fun a => wx (fX a) * wy (fY a) * wz (fZ a))
(fun _ => mul_pos (mul_pos (hx _) (hy _)) (hz _))).mass := by
let weight : A → ℝ := fun a => wx (fX a) * wy (fY a) * wz (fZ a)
have hweight : ∀ a, 0 < weight a := fun a => mul_pos (mul_pos (hx _) (hy _)) (hz _)
let p := normalizedPositiveLaw weight hweight
have htotal : 0 < ∑ a, weight a :=
Finset.sum_pos (fun a _ => hweight a) Finset.univ_nonempty
apply entropy_le_of_three_marginals p q fX fY fZ
(fun x => Real.log (wx x)) (fun y => Real.log (wy y))
(fun z => Real.log (wz z)) (Real.log (∑ a, weight a))
(normalizedPositiveLaw_mass_pos weight hweight)
· intro a
change Real.log (weight a / ∑ b, weight b) = _
rw [Real.log_div (hweight a).ne' htotal.ne']
dsimp only [weight]
rw [Real.log_mul (mul_pos (hx _) (hy _)).ne' (hz _).ne',
Real.log_mul (hx _).ne' (hy _).ne']
· exact hX
· exact hY
· exact hZ
end MatrixMultiplication.JointEntropyMax
end
end OAI
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