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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.StationaryExistence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
namespace SphericalPerceptronFreeEnergy
lemma Trial.ext_fun {m n : Trial} (h : ∀ t, m t=n t) : m=n := by
cases m
cases n
congr 1
exact funext h
lemma weightedStepTrial_coupling_fst {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (w : I → ℝ) (q : I → Time)
(hc : ∀ a, 0 ≤ c a) (hc1 : ∑ a, c a=1)
(hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1)
(hrow : ∀ i, ∑ j, c (i,j)=w i) :
weightedStepTrial c (fun a => q a.1) hc hc1=weightedStepTrial w q hw hw1 := by
apply Trial.ext_fun
intro t
change (∑ a : I×J, if q a.1 ≤ t then c a else 0)=∑ i, if q i ≤ t then w i else 0
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro i _
by_cases hi : q i ≤ t <;> simp [hi,hrow]
lemma weightedStepTrial_coupling_snd {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (v : J → ℝ) (p : J → Time)
(hc : ∀ a, 0 ≤ c a) (hc1 : ∑ a, c a=1)
(hv : ∀ i, 0 ≤ v i) (hv1 : ∑ i, v i=1)
(hcol : ∀ j, ∑ i, c (i,j)=v j) :
weightedStepTrial c (fun a => p a.2) hc hc1=weightedStepTrial v p hv hv1 := by
apply Trial.ext_fun
intro t
change (∑ a : I×J, if p a.2 ≤ t then c a else 0)=∑ j, if p j ≤ t then v j else 0
rw [Fintype.sum_prod_type,Finset.sum_comm]
apply Finset.sum_congr rfl
intro j _
by_cases hj : p j ≤ t <;> simp [hj,hcol]
lemma coupling_sum_fst {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (w f : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) :
(∑ a, c a*f a.1)=∑ i, w i*f i := by
rw [Fintype.sum_prod_type]
simp_rw [← Finset.sum_mul,hrow]
lemma coupling_sum_snd {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (v f : J → ℝ) (hcol : ∀ j, ∑ i, c (i,j)=v j) :
(∑ a, c a*f a.2)=∑ j, v j*f j := by
rw [Fintype.sum_prod_type,Finset.sum_comm]
simp_rw [← Finset.sum_mul,hcol]
lemma weightedStepTail_coupling_fst {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (w q : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) :
weightedStepTail c (fun a => q a.1)=weightedStepTail w q := by
funext t
exact coupling_sum_fst c w (fun i => 1-max (q i) t) hrow
lemma weightedStepA_coupling_fst {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (w q : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) :
weightedStepA c (fun a => q a.1)=weightedStepA w q := by
unfold weightedStepA
rw [weightedStepTail_coupling_fst c w q hrow]
lemma entropy_coupled_stationary_comparison {I J : Type*} [Fintype I] [Fintype J]
(c : I×J → ℝ) (w h : I → ℝ) (v g : J → ℝ) (q : I → Time) (p : J → Time)
(hc : ∀ a, 0 ≤ c a) (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1)
(hv : ∀ j, 0 ≤ v j) (hv1 : ∑ j, v j=1)
(hrow : ∀ i, ∑ j, c (i,j)=w i) (hcol : ∀ j, ∑ i, c (i,j)=v j)
(B : ℝ) (hB0 : 0 ≤ B) (hB1 : B < 1)
(hq : ∀ i, (q i:ℝ) ≤ B) (hp : ∀ j, (p j:ℝ) ≤ B)
(hstat : ∀ i, 2*h i=weightedStepA w (fun i => (q i:ℝ)) (q i)) :
(entropy (weightedStepTrial w q hw hw1)).toReal-∑ i, w i*h i*(q i:ℝ) ≤
(entropy (weightedStepTrial v p hv hv1)).toReal-∑ j, v j*g j*(p j:ℝ)+
∑ a, c a*|h a.1-g a.2| := by
have hc1 : ∑ a, c a=1 := by rw [Fintype.sum_prod_type]; simp_rw [hrow]; exact hw1
have hs := entropy_weightedStep_support c (fun a => p a.2) (fun a => q a.1)
hc hc1 B hB0 hB1 (fun a => hp a.2) (fun a => hq a.1)
rw [weightedStepTrial_coupling_fst c w q hc hc1 hw hw1 hrow,
weightedStepTrial_coupling_snd c v p hc hc1 hv hv1 hcol,
weightedStepA_coupling_fst c w (fun i => (q i:ℝ)) hrow] at hs
have he : (∑ a, c a*((p a.2:ℝ)-(q a.1:ℝ))*weightedStepA w (fun i => (q i:ℝ)) (q a.1))/2 =
(∑ a, c a*h a.1*(p a.2:ℝ))-(∑ i, w i*h i*(q i:ℝ)) := by
simp only [mul_assoc]
rw [← coupling_sum_fst c w (fun i => h i*(q i:ℝ)) hrow]
simp only [← mul_assoc]
rw [Finset.sum_div,← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro a _
rw [← hstat a.1]
ring
rw [he] at hs
have hb : (∑ j, v j*g j*(p j:ℝ))-(∑ a, c a*h a.1*(p a.2:ℝ)) ≤
∑ a, c a*|h a.1-g a.2| := by
simp only [mul_assoc]
rw [← coupling_sum_snd c v (fun j => g j*(p j:ℝ)) hcol]
simp only [← mul_assoc]
rw [← Finset.sum_sub_distrib]
apply Finset.sum_le_sum
intro a _
have h₁ : (g a.2-h a.1)*(p a.2:ℝ) ≤ |h a.1-g a.2| := calc
_ ≤ |g a.2-h a.1| *(p a.2:ℝ) := mul_le_mul_of_nonneg_right (le_abs_self _) (p a.2).2.1
_ ≤ |g a.2-h a.1| *1 := mul_le_mul_of_nonneg_left (p a.2).2.2 (abs_nonneg _)
_ = _ := by rw [mul_one,abs_sub_comm]
have H := mul_le_mul_of_nonneg_left h₁ (hc a)
nlinarith [H]
linarith
lemma finiteSphericalDual_eq_stationary {k : ℕ} (w h : Fin (k+1) → ℝ)
(q : Fin (k+1) → Time) (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1)
(hq : Monotone q) (hq1 : (q (Fin.last k):ℝ) < 1)
(hs : ∀ i, 2*h i=weightedStepA w (fun j => (q j:ℝ)) (q i)) :
sInf (finiteSphericalDualValues w h hw hw1)=finiteSphericalDualObjective w h hw hw1 q := by
apply IsLeast.csInf_eq
refine ⟨⟨q,⟨hq,hq1⟩,rfl⟩,?_⟩
rintro a ⟨p,⟨hp,hp1⟩,rfl⟩
exact stationary_finite_entropy_minimizes w h q p hw hw1 hq hp hq1 hp1 hs
theorem finiteSphericalDual_abs_sub_le {k : ℕ} (w h g : Fin (k+1) → ℝ)
(hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i=1)
(hh : Monotone h) (hh0 : 0 ≤ h 0) (hg : Monotone g) (hg0 : 0 ≤ g 0) :
|sInf (finiteSphericalDualValues w h (fun i => (hw i).le) hw1)-
sInf (finiteSphericalDualValues w g (fun i => (hw i).le) hw1)| ≤
∑ i, w i*|h i-g i| := by
classical
obtain ⟨q,hq,hq1,hsq⟩ := exists_finite_stationary_quantile w h hw hw1 hh hh0
obtain ⟨p,hp,hp1,hsp⟩ := exists_finite_stationary_quantile w g hw hw1 hg hg0
rw [finiteSphericalDual_eq_stationary w h q (fun i => (hw i).le) hw1 hq hq1 hsq,
finiteSphericalDual_eq_stationary w g p (fun i => (hw i).le) hw1 hp hp1 hsp]
let c : Fin (k+1)×Fin (k+1) → ℝ := fun a => if a.1=a.2 then w a.1 else 0
have hc : ∀ a, 0 ≤ c a := by intro a; dsimp [c]; split_ifs; exact (hw a.1).le; rfl
have hr : ∀ i, ∑ j, c (i,j)=w i := by intro i; simp [c]
have he : ∀ j, ∑ i, c (i,j)=w j := by intro j; simp [c]
let B := max (q (Fin.last k):ℝ) (p (Fin.last k):ℝ)
have hB0 : 0 ≤ B := (q (Fin.last k)).2.1.trans (le_max_left _ _)
have hB1 : B < 1 := max_lt hq1 hp1
have hQ (i : Fin (k+1)) : (q i:ℝ) ≤ B :=
(show (q i:ℝ) ≤ (q (Fin.last k):ℝ) from hq (Fin.le_last i)).trans (le_max_left _ _)
have hP (i : Fin (k+1)) : (p i:ℝ) ≤ B :=
(show (p i:ℝ) ≤ (p (Fin.last k):ℝ) from hp (Fin.le_last i)).trans (le_max_right _ _)
have H := entropy_coupled_stationary_comparison c w h w g q p hc
(fun i => (hw i).le) hw1 (fun i => (hw i).le) hw1 hr he B hB0 hB1 hQ hP hsq
have H' := entropy_coupled_stationary_comparison c w g w h p q hc
(fun i => (hw i).le) hw1 (fun i => (hw i).le) hw1 hr he B hB0 hB1 hP hQ hsp
have hsum : (∑ a, c a*|h a.1-g a.2|)=∑ i, w i*|h i-g i| := by
simp only [Fintype.sum_prod_type,c,ite_mul,zero_mul]
simp
have hsum' : (∑ a, c a*|g a.1-h a.2|)=∑ i, w i*|h i-g i| := by
simp only [Fintype.sum_prod_type,c,ite_mul,zero_mul]
simp [abs_sub_comm]
rw [hsum] at H
rw [hsum'] at H'
unfold finiteSphericalDualObjective
exact abs_le.mpr ⟨by linarith,by linarith⟩
end SphericalPerceptronFreeEnergy
end
end OAI
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