File size: 8,900 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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.EntropyMax
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
open scoped BigOperators Topology
open Filter MatrixMultiplication.Foundation
open MatrixMultiplication.JointEntropyMax
namespace MatrixMultiplication.JointGibbsApproximation
def marginalError {A B : Type*} [Fintype A] [Fintype B]
(p q : FiniteLaw A) (f : A → B) (v : B → ℝ) : ℝ :=
∑ b, |(q.map f).mass b - (p.map f).mass b| * |v b|
theorem marginalError_nonneg {A B : Type*} [Fintype A] [Fintype B]
(p q : FiniteLaw A) (f : A → B) (v : B → ℝ) :
0 ≤ marginalError p q f v :=
Finset.sum_nonneg fun _ _ => mul_nonneg (abs_nonneg _) (abs_nonneg _)
theorem entropy_le_crossEntropy_of_support {A : Type*} [Fintype A]
(p q : FiniteLaw A) (hsupport : ∀ a, p.mass a = 0 → q.mass a = 0) :
finiteEntropy q.mass ≤ ∑ a, -q.mass a * Real.log (p.mass a) := by
have hterm (a : A) :
entropyTerm (q.mass a) ≤ -q.mass a * Real.log (p.mass a) + p.mass a - q.mass a := by
by_cases hp : p.mass a = 0
· simp [hp, hsupport a hp]
· exact entropyTerm_le_crossEntropyTerm
(lt_of_le_of_ne (p.nonneg a) (Ne.symm hp)) (q.nonneg a)
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 _ => hterm 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
private theorem potential_difference_le {A B : Type*} [Fintype A] [Fintype B]
(p q : FiniteLaw A) (f : A → B) (v : B → ℝ) :
(∑ a, p.mass a * v (f a)) - (∑ a, q.mass a * v (f a)) ≤
marginalError p q f v := by
rw [sum_mass_mul_coordinate p f v, sum_mass_mul_coordinate q f v,
← Finset.sum_sub_distrib]
apply Finset.sum_le_sum
intro b _
calc
(p.map f).mass b * v b - (q.map f).mass b * v b =
((p.map f).mass b - (q.map f).mass b) * v b := by ring
_ ≤ |((p.map f).mass b - (q.map f).mass b) * v b| := le_abs_self _
_ = |(q.map f).mass b - (p.map f).mass b| * |v b| := by
rw [abs_mul, abs_sub_comm]
theorem entropy_le_of_marginal_errors_of_support
{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)
(hsupport : ∀ a, p.mass a = 0 → q.mass a = 0)
(hlog : ∀ a, 0 < p.mass a →
Real.log (p.mass a) = u (fX a) + v (fY a) + w (fZ a) - c) :
finiteEntropy q.mass ≤ finiteEntropy p.mass +
marginalError p q fX u + marginalError p q fY v + marginalError p q fZ w := by
have hexpand (r : FiniteLaw A) (hr : ∀ a, p.mass a = 0 → r.mass a = 0) :
(∑ a, r.mass a * Real.log (p.mass a)) =
(∑ a, r.mass a * u (fX a)) + (∑ a, r.mass a * v (fY a)) +
(∑ a, r.mass a * w (fZ a)) - c := by
have hterm (a : A) : r.mass a * Real.log (p.mass a) =
r.mass a * (u (fX a) + v (fY a) + w (fZ a) - c) := by
by_cases hp : p.mass a = 0
· simp [hr a hp]
· rw [hlog a (lt_of_le_of_ne (p.nonneg a) (Ne.symm hp))]
simp_rw [hterm, 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 hcross : (∑ a, -q.mass a * Real.log (p.mass a)) =
finiteEntropy p.mass +
((∑ a, p.mass a * u (fX a)) - (∑ a, q.mass a * u (fX a))) +
((∑ a, p.mass a * v (fY a)) - (∑ a, q.mass a * v (fY a))) +
((∑ a, p.mass a * w (fZ a)) - (∑ a, q.mass a * w (fZ a))) := by
have hp : finiteEntropy p.mass = -(∑ a, p.mass a * Real.log (p.mass a)) := by
simp only [finiteEntropy, entropyTerm, neg_mul, Finset.sum_neg_distrib]
simp only [neg_mul, Finset.sum_neg_distrib]
rw [hp, hexpand p (fun _ h => h), hexpand q hsupport]
ring
calc
finiteEntropy q.mass ≤ ∑ a, -q.mass a * Real.log (p.mass a) :=
entropy_le_crossEntropy_of_support p q hsupport
_ = _ := hcross
_ ≤ _ := add_le_add
(add_le_add (add_le_add_right (potential_difference_le p q fX u) _)
(potential_difference_le p q fY v)) (potential_difference_le p q fZ w)
theorem entropy_le_of_marginal_errors
{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) :
finiteEntropy q.mass ≤ finiteEntropy p.mass +
marginalError p q fX u + marginalError p q fY v + marginalError p q fZ w :=
entropy_le_of_marginal_errors_of_support p q fX fY fZ u v w c
(fun a h => ((hp a).ne' h).elim) (fun a _ => hlog a)
theorem tendsto_map_mass {I A B : Type*} [Fintype A] [Fintype B]
{l : Filter I} {pN : I → FiniteLaw A} {p : FiniteLaw A}
(h : ∀ a, Tendsto (fun n => (pN n).mass a) l (𝓝 (p.mass a)))
(f : A → B) (b : B) :
Tendsto (fun n => ((pN n).map f).mass b) l (𝓝 ((p.map f).mass b)) := by
classical
simp_rw [FiniteLaw.map_mass]
apply tendsto_finsetSum
intro a _
by_cases hab : f a = b
· simpa only [hab, ite_true] using h a
· simp only [hab, ite_false]
exact tendsto_const_nhds
theorem tendsto_marginalError {I A B : Type*} [Fintype A] [Fintype B]
{l : Filter I} {pN : I → FiniteLaw A} {p : FiniteLaw A}
(h : ∀ a, Tendsto (fun n => (pN n).mass a) l (𝓝 (p.mass a)))
(f : A → B) (v : B → ℝ) :
Tendsto (fun n => marginalError p (pN n) f v) l (𝓝 0) := by
have hterm (b : B) :
Tendsto (fun n => |((pN n).map f).mass b - (p.map f).mass b| * |v b|)
l (𝓝 0) := by
have hconst : Tendsto (fun _ : I => (p.map f).mass b) l
(𝓝 ((p.map f).mass b)) := tendsto_const_nhds
simpa only [sub_self, abs_zero, zero_mul] using
((tendsto_map_mass h f b).sub hconst).abs.mul_const |v b|
simpa only [marginalError, Finset.sum_const_zero] using
tendsto_finsetSum Finset.univ (fun b _ => hterm b)
theorem eventually_entropy_le_of_tendsto_of_support
{I A X Y Z : Type*} [Fintype A] [Fintype X] [Fintype Y] [Fintype Z]
{l : Filter I} (p : FiniteLaw A) (pN : I → FiniteLaw A)
(fX : A → X) (fY : A → Y) (fZ : A → Z)
(u : X → ℝ) (v : Y → ℝ) (w : Z → ℝ) (c : ℝ)
(hlog : ∀ a, 0 < p.mass a →
Real.log (p.mass a) = u (fX a) + v (fY a) + w (fZ a) - c)
(hN : ∀ a, Tendsto (fun n => (pN n).mass a) l (𝓝 (p.mass a)))
{η : ℝ} (hη : 0 < η) :
∀ᶠ n in l, ∀ q : FiniteLaw A,
(∀ a, p.mass a = 0 → q.mass a = 0) →
(q.map fX).mass = ((pN n).map fX).mass →
(q.map fY).mass = ((pN n).map fY).mass →
(q.map fZ).mass = ((pN n).map fZ).mass →
finiteEntropy q.mass ≤ finiteEntropy p.mass + η := by
have herror : Tendsto (fun n => marginalError p (pN n) fX u +
marginalError p (pN n) fY v + marginalError p (pN n) fZ w) l (𝓝 0) := by
simpa only [add_zero] using
((tendsto_marginalError hN fX u).add (tendsto_marginalError hN fY v)).add
(tendsto_marginalError hN fZ w)
have hsmall := herror.eventually (gt_mem_nhds hη)
filter_upwards [hsmall] with n hn
intro q hs hX hY hZ
have hbound := entropy_le_of_marginal_errors_of_support p q fX fY fZ u v w c hs hlog
have hx : marginalError p q fX u = marginalError p (pN n) fX u := by
unfold marginalError
rw [hX]
have hy : marginalError p q fY v = marginalError p (pN n) fY v := by
unfold marginalError
rw [hY]
have hz : marginalError p q fZ w = marginalError p (pN n) fZ w := by
unfold marginalError
rw [hZ]
rw [hx, hy, hz] at hbound
linarith
theorem eventually_entropy_le_of_tendsto
{I A X Y Z : Type*} [Fintype A] [Fintype X] [Fintype Y] [Fintype Z]
{l : Filter I} (p : FiniteLaw A) (pN : I → 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)
(hN : ∀ a, Tendsto (fun n => (pN n).mass a) l (𝓝 (p.mass a)))
{η : ℝ} (hη : 0 < η) :
∀ᶠ n in l, ∀ q : FiniteLaw A,
(q.map fX).mass = ((pN n).map fX).mass →
(q.map fY).mass = ((pN n).map fY).mass →
(q.map fZ).mass = ((pN n).map fZ).mass →
finiteEntropy q.mass ≤ finiteEntropy p.mass + η := by
filter_upwards [eventually_entropy_le_of_tendsto_of_support p pN fX fY fZ u v w c
(fun a _ => hlog a) hN hη] with n hn
intro q hX hY hZ
exact hn q (fun a h => ((hp a).ne' h).elim) hX hY hZ
end MatrixMultiplication.JointGibbsApproximation
end
end OAI
|